The semi-classical limit of large fermionic systems

  • Søren Fournais
  • Mathieu Lewin
  • Jan Philip Solovej


We study a system of N fermions in the regime where the intensity of the interaction scales as 1 / N and with an effective semi-classical parameter \(\hbar =N^{-1/d}\) where d is the space dimension. For a large class of interaction potentials and of external electromagnetic fields, we prove the convergence to the Thomas–Fermi minimizers in the limit \(N\rightarrow \infty \). The limit is expressed using many-particle coherent states and Wigner functions. The method of proof is based on a fermionic de Finetti–Hewitt–Savage theorem in phase space and on a careful analysis of the possible lack of compactness at infinity.

Mathematics Subject Classification




M.L. and J.P.S acknowledge financial support from the European Research Council (Grant Agreements MNIQS 258023 and MASTRUMAT 321029). S.F. acknowledges support from a Danish research council Sapere Aude grant. This work was started when the authors were at the Centre Émile Borel of the Institut Henri Poincaré in Paris in 2013. Part of this work was done when S.F. was a visiting professor at the University Paris-Dauphine.


  1. 1.
    Ammari, Z., Nier, F.: Mean field limit for bosons and infinite dimensional phase-space analysis. Ann. Henri Poincaré 9, 1503–1574 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Auchmuty, J.F.G., Beals, R.: Models of rotating stars. Astrophys. J. 165, L79+ (1971)CrossRefGoogle Scholar
  3. 3.
    Auchmuty, J.F.G., Beals, R.: Variational solutions of some nonlinear free boundary problems. Arch. Ration. Mech. Anal. 43, 255–271 (1971)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Bach, V.: Ionization energies of bosonic Coulomb systems. Lett. Math. Phys. 21, 139–149 (1991)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Bach, V., Breteaux, S., Petrat, S., Pickl, P., Tzaneteas, T.: Kinetic energy estimates for the accuracy of the time-dependent Hartree-Fock approximation with coulomb interaction. J. Math. Pures Appl. 105, 1–30 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Bach, V., Lewis, R., Lieb, E.H., Siedentop, H.: On the number of bound states of a bosonic \(N\)-particle Coulomb system. Math. Z. 214, 441–459 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Bardos, C., Golse, F., Gottlieb, A.D., Mauser, N.J.: Mean field dynamics of fermions and the time-dependent Hartree–Fock equation. J. Math. Pures Appl. (9) 82, 665–683 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Bardos, C., Golse, F., Mauser, N.J.: Weak coupling limit of the \(N\)-particle Schrödinger equation. Methods Appl. Anal. 7, 275–293 (2000). Cathleen Morawetz: a great mathematicianMathSciNetzbMATHGoogle Scholar
  9. 9.
    Benedikter, N., Jaksic, V., Porta, M., Saffirio, C., Schlein, B.: Mean-field evolution of Fermionic mixed states. Commun. Pure Appl. Math. 69, 2250–2303 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Benedikter, N., Porta, M., Saffirio, C., Schlein, B.: From the Hartree dynamics to the Vlasov equation. Arch. Ration. Mech. Anal. 221(1), 273–334 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Benedikter, N., Porta, M., Schlein, B.: Mean-field evolution of fermionic systems. Commun. Math. Phys. 331, 1087–1131 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Benguria, R., Lieb, E.H.: Proof of the stability of highly negative ions in the absence of the Pauli principle. Phys. Rev. Lett. 50, 1771–1774 (1983)CrossRefGoogle Scholar
  13. 13.
    Caglioti, E., Lions, P.-L., Marchioro, C., Pulvirenti, M.: A special class of stationary flows for two-dimensional Euler equations: a statistical mechanics description. Commun. Math. Phys. 143, 501–525 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Combescure, M., Robert, D.: Coherent States and Applications in Mathematical Physics, Theoretical and Mathematical Physics. Springer, Dordrecht (2012)zbMATHGoogle Scholar
  15. 15.
    de Finetti, B.: Funzione caratteristica di un fenomeno aleatorio. Atti della R. Accademia Nazionale dei Lincei. Ser. 6, Memorie, Classe di Scienze Fisiche, Matematiche e Naturali (1931)Google Scholar
  16. 16.
    de Finetti, B.: La prévision: ses lois logiques, ses sources subjectives. Ann. Inst. H. Poincaré 7, 1–68 (1937)MathSciNetzbMATHGoogle Scholar
  17. 17.
    Dereziński, J., Gérard, C.: Mathematics of Quantization and Quantum Fields, Cambridge Monographs on Mathematical Physics. Cambridge University Press, Cambridge (2013)CrossRefzbMATHGoogle Scholar
  18. 18.
    Diaconis, P., Freedman, D.: Finite exchangeable sequences. Ann. Prob. 8, 745–764 (1980)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Dynkin, E .B.: Classes of equivalent random quantities. Uspehi Matem. Nauk (N.S.) 8, 125–130 (1953)MathSciNetGoogle Scholar
  20. 20.
    Dyson, F.J., Lenard, A.: Stability of matter. I. J. Math. Phys. 8, 423–434 (1967)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Elgart, A., Erdős, L., Schlein, B., Yau, H.-T.: Nonlinear Hartree equation as the mean field limit of weakly coupled fermions. J. Math. Pures Appl. 83, 1241–1273 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Elgart, A., Erdős, L., Schlein, B., Yau, H.-T.: Gross-Pitaevskii equation as the mean field limit of weakly coupled bosons. Arch. Ration. Mech. Anal. 179, 265–283 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Elgart, A., Schlein, B.: Mean field dynamics of boson stars. Commun. Pure Appl. Math. 60, 500–545 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Elliott, P., Lee, D., Cangi, A., Burke, K.: Semiclassical origins of density functionals. Phys. Rev. Lett. 100, 256406 (2008)CrossRefGoogle Scholar
  25. 25.
    Erdös, L., Schlein, B., Yau, H.-T.: Rigorous derivation of the Gross–Pitaevskii equation with a large interaction potential. J. Am. Math. Soc. 22, 1099–1156 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Fannes, M., Spohn, H., Verbeure, A.: Equilibrium states for mean field models. J. Math. Phys. 21, 355–358 (1980)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Friedman, A.: Variational Principles and Free-Boundary Problems, Pure and Applied Mathematics. Wiley, New York (1982). A Wiley-Interscience PublicationGoogle Scholar
  28. 28.
    Fröhlich, J., Knowles, A.: A microscopic derivation of the time-dependent Hartree–Fock equation with Coulomb two-body interaction. J. Stat. Phys. 145, 23–50 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Fröhlich, J., Knowles, A., Schwarz, S.: On the mean-field limit of bosons with Coulomb two-body interaction. Commun. Math. Phys. 288, 1023–1059 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Fröhlich, J., Graffi, S., Schwarz, S.: Mean-field and classical limit of many-body Schrödinger dynamics for bosons. Commun. Math. Phys. 271, 681–697 (2007)CrossRefzbMATHGoogle Scholar
  31. 31.
    Ginibre, J., Velo, G.: The classical field limit of scattering theory for nonrelativistic many-boson systems. I. Commun. Math. Phys. 66, 37–76 (1979)MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    Golse, F.: On the dynamics of large particle systems in the mean field limit, ArXiv e-prints arXiv:1301.5494. Lecture notes for a course at the NDNS+ Applied Dynamical Systems Summer School “Macroscopic and large scale phenomena”. Universiteit Twente, Enschede (The Netherlands) (2013)
  33. 33.
    Graffi, S., Martinez, A., Pulvirenti, M.: Mean-field approximation of quantum systems and classical limit. Math. Methods Appl. Sci. 13, 59–73 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  34. 34.
    Grech, P., Seiringer, R.: The excitation spectrum for weakly interacting bosons in a trap. Commun. Math. Phys. 322, 559–591 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  35. 35.
    Hepp, K.: The classical limit for quantum mechanical correlation functions. Commun. Math. Phys. 35, 265–277 (1974)MathSciNetCrossRefGoogle Scholar
  36. 36.
    Hewitt, E., Savage, L.J.: Symmetric measures on Cartesian products. Trans. Am. Math. Soc. 80, 470–501 (1955)MathSciNetCrossRefzbMATHGoogle Scholar
  37. 37.
    Husimi, K.: Some formal properties of the density matrix. Proc. Phys. Math. Soc. Jpn. 22, 264 (1940)zbMATHGoogle Scholar
  38. 38.
    Hwang, I.: The \(L^2\)-boundedness of pseudo differential operators. Trans. Am. Math. Soc 302, 55–76 (1987)Google Scholar
  39. 39.
    Kiessling, M.K.-H.: The Hartree limit of Born’s ensemble for the ground state of a bosonic atom or ion. J. Math. Phys. 53, 095223 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  40. 40.
    Knowles, A., Pickl, P.: Mean-field dynamics: singular potentials and rate of convergence. Commun. Math. Phys. 298, 101–138 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  41. 41.
    Lévy-Leblond, J.-M.: Nonsaturation of gravitational forces. J. Math. Phys. 10, 806–812 (1969)CrossRefzbMATHGoogle Scholar
  42. 42.
    Lewin, M.: Geometric methods for nonlinear many-body quantum systems. J. Funct. Anal. 260, 3535–3595 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  43. 43.
    Lewin, M.: Mean-field limit of Bose systems: rigorous results. In: Proceedings of the International Congress of Mathematical Physics (2015). ArXiv e-printsGoogle Scholar
  44. 44.
    Lewin, M., Nam, P.T., Rougerie, N.: Derivation of Hartree’s theory for generic mean-field Bose systems. Adv. Math. 254, 570–621 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  45. 45.
    Lewin, M., Nam, P.T., Rougerie, N.: Remarks on the quantum de Finetti theorem for bosonic systems. Appl. Math. Res. Express (AMRX) 2015, 48–63 (2015)MathSciNetzbMATHGoogle Scholar
  46. 46.
    Lewin, M., Nam, P.T., Rougerie, N.: The mean-field approximation and the non-linear Schrödinger functional for trapped Bose gases. Trans. Am. Math. Soc 368, 6131–6157 (2016)CrossRefzbMATHGoogle Scholar
  47. 47.
    Lewin, M., Nam, P.T., Schlein, B.: Fluctuations around Hartree states in the mean-field regime. Am. J. Math. 137, 1613–1650 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  48. 48.
    Lewin, M., Thành Nam, P., Rougerie, N.: A note on 2D focusing many-boson systems. Proc. Am. Math. Soc. 145, 2441–2454 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  49. 49.
    Lieb, E.H.: Thomas–Fermi and related theories of atoms and molecules. Rev. Mod. Phys. 53, 603–641 (1981)MathSciNetCrossRefzbMATHGoogle Scholar
  50. 50.
    Lieb, E .H., Liniger, W.: Exact analysis of an interacting Bose gas. I. The general solution and the ground state. Phys. Rev. (2) 130, 1605–1616 (1963)MathSciNetCrossRefzbMATHGoogle Scholar
  51. 51.
    Lieb, E.H., Seiringer, R.: Derivation of the Gross–Pitaevskii equation for rotating Bose gases. Commun. Math. Phys. 264, 505–537 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  52. 52.
    Lieb, E.H., Seiringer, R.: The Stability of Matter in Quantum Mechanics. Cambridge University Press, Cambridge (2010)zbMATHGoogle Scholar
  53. 53.
    Lieb, E.H., Seiringer, R., Solovej, J.P., Yngvason, J.: The Mathematics of the Bose Gas and its Condensation. Oberwolfach Seminars, Birkhäuser (2005)zbMATHGoogle Scholar
  54. 54.
    Lieb, E.H., Simon, B.: Thomas–Fermi theory revisited. Phys. Rev. Lett. 31, 681–683 (1973)CrossRefGoogle Scholar
  55. 55.
    Lieb, E.H., Simon, B.: The Hartree–Fock theory for Coulomb systems. Commun. Math. Phys. 53, 185–194 (1977)MathSciNetCrossRefGoogle Scholar
  56. 56.
    Lieb, E.H., Simon, B.: The Thomas–Fermi theory of atoms, molecules and solids. Adv. Math. 23, 22–116 (1977)MathSciNetCrossRefzbMATHGoogle Scholar
  57. 57.
    Lieb, E.H., Thirring, W.E.: Bound on kinetic energy of fermions which proves stability of matter. Phys. Rev. Lett. 35, 687–689 (1975)CrossRefGoogle Scholar
  58. 58.
    Lieb, E.H., Thirring, W.E.: Inequalities for the Moments of the Eigenvalues of the Schrödinger Hamiltonian and their Relation to Sobolev Inequalities, Studies in Mathematical Physics, pp. 269–303. Princeton University Press, Princeton (1976)Google Scholar
  59. 59.
    Lieb, E.H., Thirring, W.E.: Gravitational collapse in quantum mechanics with relativistic kinetic energy. Ann. Phys. 155, 494–512 (1984)MathSciNetCrossRefGoogle Scholar
  60. 60.
    Lieb, E.H., Yau, H.-T.: The Chandrasekhar theory of stellar collapse as the limit of quantum mechanics. Commun. Math. Phys. 112, 147–174 (1987)MathSciNetCrossRefzbMATHGoogle Scholar
  61. 61.
    Lions, P.-L.: Minimization problems in \(L^{1}({ R}^{3})\). J. Funct. Anal. 41, 236–275 (1981)MathSciNetCrossRefGoogle Scholar
  62. 62.
    Lions, P.-L.: The concentration-compactness principle in the calculus of variations. The locally compact case, Part I. Ann. Inst. H. Poincaré Anal. Non Linéaire 1, 109–149 (1984)MathSciNetCrossRefzbMATHGoogle Scholar
  63. 63.
    Lions, P.-L.: Solutions of Hartree–Fock equations for Coulomb systems. Commun. Math. Phys. 109, 33–97 (1987)MathSciNetCrossRefzbMATHGoogle Scholar
  64. 64.
    Lions, P.-L., Paul, T.: Sur les mesures de Wigner. Rev. Mat. Iberoamericana 9, 553–618 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  65. 65.
    Messer, J., Spohn, H.: Statistical mechanics of the isothermal Lane–Emden equation. J. Stat. Phys. 29, 561–578 (1982)MathSciNetCrossRefGoogle Scholar
  66. 66.
    Narnhofer, H., Sewell, G.: Vlasov hydrodynamics of a quantum mechanical model. Commun. Math. Phys. 79, 9–24 (1981)MathSciNetCrossRefGoogle Scholar
  67. 67.
    Petrat, S., Pickl, P.: A new method and a new scaling for deriving Fermionic mean-field dynamics, ArXiv e-prints (2014)Google Scholar
  68. 68.
    Pickl, P.: A simple derivation of mean-field limits for quantum systems. Lett. Math. Phys. 97, 151–164 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  69. 69.
    Raggio, G.A., Werner, R.F.: Quantum statistical mechanics of general mean field systems. Helv. Phys. Acta 62, 980–1003 (1989)MathSciNetzbMATHGoogle Scholar
  70. 70.
    Rodnianski, I., Schlein, B.: Quantum fluctuations and rate of convergence towards mean field dynamics. Commun. Math. Phys. 291, 31–61 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  71. 71.
    Rougerie, N.: De Finetti theorems, mean-field limits and Bose–Einstein condensation, ArXiv e-prints (2015)Google Scholar
  72. 72.
    Seiringer, R.: The excitation spectrum for weakly interacting bosons. Commun. Math. Phys. 306, 565–578 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  73. 73.
    Seiringer, R., Yngvason, J., Zagrebnov, V.A.: Disordered Bose–Einstein condensates with interaction in one dimension. J. Stat. Mech. 2012, P11007 (2012)CrossRefGoogle Scholar
  74. 74.
    Solovej, J.P.: Asymptotics for bosonic atoms. Lett. Math. Phys. 20, 165–172 (1990)MathSciNetCrossRefzbMATHGoogle Scholar
  75. 75.
    Solovej, J .P.: The ionization conjecture in Hartree–Fock theory. Ann. Math. (2) 158, 509–576 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  76. 76.
    Spohn, H.: Kinetic equations from Hamiltonian dynamics: Markovian limits. Rev. Modern Phys. 52, 569–615 (1980)MathSciNetCrossRefGoogle Scholar
  77. 77.
    Spohn, H.: On the Vlasov hierarchy. Math. Methods Appl. Sci. 3, 445–455 (1981)MathSciNetCrossRefzbMATHGoogle Scholar
  78. 78.
    Takahashi, K.: Wigner and Husimi functions in quantum mechanics. J. Phys. Soc. Jpn. 55, 762–779 (1986)MathSciNetCrossRefGoogle Scholar
  79. 79.
    van den Berg, M., Lewis, J.T., Pulè, J.V.: The large deviation principle and some models of an interacting boson gas. Commun. Math. Phys. 118, 61–85 (1988)MathSciNetCrossRefzbMATHGoogle Scholar
  80. 80.
    Werner, R.F.: Large deviations and mean-field quantum systems. In: Accardi, L. (ed.) Quantum Probability and Telated Topics, QP–PQ, vol. VII, pp. 349–381. World Scientific Publication, River Edge, NJ (1992)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  • Søren Fournais
    • 1
  • Mathieu Lewin
    • 2
  • Jan Philip Solovej
    • 3
  1. 1.Department of MathematicsAarhus UniversityAarhus CDenmark
  2. 2.CEREMADE, CNRS University Paris-DauphinePSL Research University Place de Lattre de TassignyParisFrance
  3. 3.Department of MathematicsUniversity of CopenhagenCopenhagen ØDenmark

Personalised recommendations