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Mean–Field Evolution of Fermionic Systems

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Abstract

The mean field limit for systems of many fermions is naturally coupled with a semiclassical limit. This makes the analysis of the mean field regime much more involved, compared with bosonic systems. In this paper, we study the dynamics of initial data close to a Slater determinant, whose reduced one-particle density is an orthogonal projection ω N with the appropriate semiclassical structure. Assuming some regularity of the interaction potential, we show that the evolution of such an initial data remains close to a Slater determinant, with reduced one-particle density given by the solution of the Hartree–Fock equation with initial data ω N . Our result holds for all (semiclassical) times, and gives effective bounds on the rate of the convergence towards the Hartree–Fock dynamics.

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References

  1. Adami R., Golse R., Teta A.: Rigorous derivation of the cubic NLS in dimension one. J. Stat. Phys. 127(6), 1193–1220 (2007)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  2. Ammari Z., Nier F.: Mean field propagation of Wigner measures and BBGKY hierarchy for general bosonic states. J. Math. Pures Appl. (9) 95(6), 585–626 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  3. Bach V.: Error bound for the Hartree–Fock energy of atoms and molecules. Commun. Math. Phys. 147(3), 527–548 (1992)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  4. 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(6), 665–683 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  5. Ben Arous G., Kirkpatrick K., Schlein B.: A central limit theorem in Many–Body quantum dynamics. Commun. math. Phys. 321(2), 371–417 (2013)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  6. Benedikter, N., de Oliveira, G., Schlein. B.: Quantitative derivation of the Gross–Pitaevskii Equation. To appear in Comm. Pure Appl. Math. (2014). arxiv:1208.0373 [math-ph]

  7. Chen L., Oon Lee J., Schlein B.: Rate of convergence towards Hartree dynamics. J. Stat. Phys. 144(4), 872–903 (2011)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  8. 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. (9) 83(10), 1241–1273 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  9. Erdős L., Yau H.-T.: Derivation of the nonlinear Schrödinger equation from a many body Coulomb system. Adv. Theor. Math. Phys. 5(6), 1169–1205 (2001)

    MathSciNet  Google Scholar 

  10. Erdős L., Schlein B., Yau H.-T.: Derivation of the cubic non-linear Schrödinger equation from quantum dynamics of many-body systems. Inv. Math. 167, 515–614 (2006)

    Article  ADS  Google Scholar 

  11. Erdős L., Schlein B., Yau H.-T.: Derivation of the Gross–Pitaevskii equation for the dynamics of Bose–Einstein condensate. Ann. Math. (2) 172(1), 291–370 (2010)

    Article  Google Scholar 

  12. Erdős L., Schlein B., Yau H.-T.: Rigorous derivation of the Gross–Pitaevskii equation. Phys. Rev. Lett. 98(4), 040404 (2007)

    Article  ADS  Google Scholar 

  13. Erdős L., Schlein B., Yau H.-T.: Rigorous derivation of the Gross–Pitaevskii equation with a large interaction potential. J. Amer. Math. Soc. 22, 1099–1156 (2009)

    Article  MathSciNet  Google Scholar 

  14. Frank R.L., Hainzl C., Seiringer R., Solovej J.P.: Microscopic derivation of Ginzburg–Landau theory. J. Amer. Math. Soc. 25, 667–713 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  15. Fröhlich J., Graffi S., Schwarz S.: Mean–field- and classical limit of many-body Schrödinger dynamics for bosons. Commun. Math. Phys. 271(3), 681–697 (2007)

    Article  ADS  MATH  Google Scholar 

  16. Fröhlich J., Knowles A.: A microscopic derivation of the time-dependent Hartree-Fock equation with Coulomb two-body interaction. J. Stat. Phys. 145(1), 23–50 (2011)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  17. Fröhlich J., Knowles A., Schwarz S.: On the mean–field limit of bosons with Coulomb two-body interaction. Commun. Math. Phys. 288(3), 1023–1059 (2009)

    Article  ADS  MATH  Google Scholar 

  18. Ginibre J., Velo G.: The classical field limit of scattering theory for nonrelativistic many-boson systems. I and II. Commun. Math. Phys. 66(1), 37–76 (1979) 68(1):45–68 (1979)

  19. Graf G.M., Solovej J.P.: A correlation estimate with applications to quantum systems with Coulomb interactions. Rev. Math. Phys. 6, 977–997 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  20. Graffi S., Martinez A., Pulvirenti M.: Mean–field approximation of quantum systems and classical limit. Math. Models Methods Appl. Sci. 13(1), 59–73 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  21. Grillakis M., Machedon M., Margetis D.: Second-order corrections to mean field evolution of weakly interacting bosons. I. Commun. Math. Phys. 294(1), 273–301 (2010)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  22. Grillakis M., Machedon M., Margetis D.: Second-order corrections to mean field evolution of weakly interacting bosons. II. Adv. Math. 228(3), 1788–1815 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  23. Grillakis M., Machedon M.: Pair excitations and the mean field approximation of interacting bosons. I. Commun. Math. Phys. 324(2), 601–636 (2013)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  24. Grech P., Seiringer R.: The excitation spectrum for weakly interacting bosons in a trap. Commun. Math. Phys. 322(2), 559–591 (2013)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  25. Hainzl C., Hamza E., Seiringer R., Solovej J.P.: The BCS functional for general pair interactions. Commun. Math. Phys. 281, 349–367 (2008)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  26. Hainzl C., Lenzmann E., Lewin M., Schlein B.: On Blowup for time-dependent generalized Hartree–Fock equations. Ann. Henri Poincare 11, 1023 (2010)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  27. Hainzl C., Seiringer R.: Critical temperature and energy gap in the BCS equation. Phys. Rev. B 77, 184517 (2008)

    Article  ADS  Google Scholar 

  28. Hainzl C., Seiringer R.: Low density limit of BCS theory and Bose–Einstein condensation of fermion pairs. Lett. Math. Phys. 100(2), 119–138 (2012)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  29. Hainzl C., Schlein B.: Dynamics of Bose–Einstein condensates of fermion pairs in the low density limit of BCS theory. J. Funct. Anal. 265(3), 399–423 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  30. Hepp K.: The classical limit for quantum mechanical correlation functions. Commun. Math. Phys. 35, 265–277 (1974)

    Article  ADS  MathSciNet  Google Scholar 

  31. Kirkpatrick K., Schlein B., Staffilani G.: Derivation of the two dimensional nonlinear Schrödinger equation from many body quantum dynamics. Amer. J. Math. 133(1), 91–130 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  32. Knowles A., Pickl P.: Mean–field dynamics: singular potentials and rate of convergence. Commun. Math. Phys. 298(1), 101–138 (2010)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  33. Lewin, M., Thành Nam, P., Serfaty, S., Solovej, J.P.: Bogoliubov spectrum of interacting Bose gases. To appear in Comm. Pure Appl. Math. (2013). arxiv:1211.2778 [math-ph]

  34. Narnhofer H., Sewell G.L.: Vlasov hydrodynamics of a quantum mechanical model. Commun. Math. Phys. 79(1), 9–24 (1981)

    Article  ADS  MathSciNet  Google Scholar 

  35. Pickl P.: A simple derivation of mean field limits for quantum systems. Lett. Math. Phys. 97(2), 151–164 (2011)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  36. Rodnianski I., Schlein B.: Quantum fluctuations and rate of convergence towards mean field dynamics. Commun. Math. Phys. 291(1), 31–61 (2009)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  37. Ruijsenaars S.N.M.: On Bogoliubov transformations. II. The general case. Ann. Phys. 116, 105–134 (1978)

    Article  ADS  MathSciNet  Google Scholar 

  38. Seiringer, R.: (2011) The excitation spectrum for weakly interacting bosons. Commun. Math. Phys. 306(2), 565–578

    Google Scholar 

  39. Solovej, J.P.: Many Body Quantum Mechanics. Lecture Notes. Summer (2007). Available at http://www.mathematik.uni-muenchen.de/~sorensen/Lehre/SoSe2013/MQM2/skript.pdf

  40. Spohn H.: Kinetic equations from Hamiltonian dynamics: Markovian limits. Rev. Modern Phys. 52(3), 569–615 (1980)

    Article  ADS  MathSciNet  Google Scholar 

  41. Spohn H.: On the Vlasov hierarchy. Math. Methods Appl. Sci. 3(4), 445–455 (1981)

    Article  ADS  MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Institute of Applied Mathematics, University of Bonn, Endenicher Allee 60, 53115, Bonn, Germany

    Niels Benedikter, Marcello Porta & Benjamin Schlein

Authors
  1. Niels Benedikter
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  2. Marcello Porta
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  3. Benjamin Schlein
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Corresponding author

Correspondence to Benjamin Schlein.

Additional information

Communicated by H. Spohn

M. Porta supported by ERC Grant MAQD 240518.

B. Schlein partially supported by ERC Grant MAQD 240518.

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Benedikter, N., Porta, M. & Schlein, B. Mean–Field Evolution of Fermionic Systems. Commun. Math. Phys. 331, 1087–1131 (2014). https://doi.org/10.1007/s00220-014-2031-z

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