Mean-field evolution of fermionic systems

Marcello Porta

Séminaire Laurent Schwartz — EDP et applications (2014-2015)

  • Volume: 331, Issue: 3, page 1-13
  • ISSN: 2266-0607

Abstract

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We study the dynamics of interacting fermionic systems, in the mean-field regime. We consider initial states which are close to quasi-free states and prove that, under suitable assumptions on the inital data and on the many-body interaction, the quantum evolution of the system is approximated by a time-dependent quasi-free state. In particular we prove that the evolution of the reduced one-particle density matrix converges, as the number of particles goes to infinity, to the solution of the time-dependent Hartree-Fock equation. Our theorems allow to describe the dynamics of both pure states (zero temperature states) and mixed states (positive temperature states). Our results hold for all times, and give effective estimates on the rate of convergence towards the Hartree-Fock evolution. The results on pure states are based on joint works with N. Benedikter and B. Schlein, [5, 6]; while those on mixed states are based on a joint work with N. Benedikter, V. Jaksic, C. Saffirio and B. Schlein, [7].

How to cite

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Porta, Marcello. "Mean-field evolution of fermionic systems." Séminaire Laurent Schwartz — EDP et applications 331.3 (2014-2015): 1-13. <http://eudml.org/doc/275807>.

@article{Porta2014-2015,
abstract = {We study the dynamics of interacting fermionic systems, in the mean-field regime. We consider initial states which are close to quasi-free states and prove that, under suitable assumptions on the inital data and on the many-body interaction, the quantum evolution of the system is approximated by a time-dependent quasi-free state. In particular we prove that the evolution of the reduced one-particle density matrix converges, as the number of particles goes to infinity, to the solution of the time-dependent Hartree-Fock equation. Our theorems allow to describe the dynamics of both pure states (zero temperature states) and mixed states (positive temperature states). Our results hold for all times, and give effective estimates on the rate of convergence towards the Hartree-Fock evolution. The results on pure states are based on joint works with N. Benedikter and B. Schlein, [5, 6]; while those on mixed states are based on a joint work with N. Benedikter, V. Jaksic, C. Saffirio and B. Schlein, [7].},
author = {Porta, Marcello},
journal = {Séminaire Laurent Schwartz — EDP et applications},
keywords = {fermionic systems; Hamiltonian; Hilbert space; Hartree dynamics; Schrödinger equation; Hartree-Fock equation; initial data; solution},
language = {eng},
number = {3},
pages = {1-13},
publisher = {Institut des hautes études scientifiques & Centre de mathématiques Laurent Schwartz, École polytechnique},
title = {Mean-field evolution of fermionic systems},
url = {http://eudml.org/doc/275807},
volume = {331},
year = {2014-2015},
}

TY - JOUR
AU - Porta, Marcello
TI - Mean-field evolution of fermionic systems
JO - Séminaire Laurent Schwartz — EDP et applications
PY - 2014-2015
PB - Institut des hautes études scientifiques & Centre de mathématiques Laurent Schwartz, École polytechnique
VL - 331
IS - 3
SP - 1
EP - 13
AB - We study the dynamics of interacting fermionic systems, in the mean-field regime. We consider initial states which are close to quasi-free states and prove that, under suitable assumptions on the inital data and on the many-body interaction, the quantum evolution of the system is approximated by a time-dependent quasi-free state. In particular we prove that the evolution of the reduced one-particle density matrix converges, as the number of particles goes to infinity, to the solution of the time-dependent Hartree-Fock equation. Our theorems allow to describe the dynamics of both pure states (zero temperature states) and mixed states (positive temperature states). Our results hold for all times, and give effective estimates on the rate of convergence towards the Hartree-Fock evolution. The results on pure states are based on joint works with N. Benedikter and B. Schlein, [5, 6]; while those on mixed states are based on a joint work with N. Benedikter, V. Jaksic, C. Saffirio and B. Schlein, [7].
LA - eng
KW - fermionic systems; Hamiltonian; Hilbert space; Hartree dynamics; Schrödinger equation; Hartree-Fock equation; initial data; solution
UR - http://eudml.org/doc/275807
ER -

References

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  1. H. Araki and W. Wyss. Representations of canonical anticommutation relations. Helv. Phys. Acta37 (1964), 136. Zbl0137.23903MR171521
  2. A. Athanassoulis, T. Paul, F. Pezzotti and M. Pulvirenti. Strong semiclassical approximation of Wigner functions for the Hartree dynamics. Atti della Accademia Nazionale dei Lincei. Rendiconti Lincei. Matematica e Applicazioni.22, 525–552 (2011). Zbl1235.81100MR2904998
  3. V. Bach. Error bound for the Hartree-Fock energy of atoms and molecules. Comm. Math. Phys.147 (1992), no. 3, 527–548. Zbl0771.46038MR1175492
  4. C. Bardos, F. Golse, A. D. Gottlieb, and N. J. Mauser. Mean field dynamics of fermions and the time-dependent Hartree-Fock equation. J. Math. Pures Appl. (9)82 (2003), no. 6, 665–683. Zbl1029.82022MR1996777
  5. N. Benedikter, M. Porta and B. Schlein. Mean-field evolution of fermionic systems. Comm. Math. Phys.331, 1087–1131 (2014). Zbl1304.82061MR3248060
  6. N. Benedikter, M. Porta and B. Schlein. Mean-field dynamics of fermions with relativistic dispersion. J. Math. Phys.55, 021901 (2014). Zbl1286.81183MR3202863
  7. N. Benedikter, V. Jakšić, M. Porta, C. Saffirio and B. Schlein. Mean-field evolution of fermionic mixed states. http://arxiv.org/abs/1411.0843 
  8. W. Braun and K. Hepp. The Vlasov dynamics and its fluctuations in the 1 / N limit of interacting classical particles. Comm. Math. Phys.56, 101–113 (1977). Zbl1155.81383MR475547
  9. J. Derezinśki and C. Gérard. Mathematics of quantization and quantum fields. Cambridge University press (2013). Zbl1271.81004MR3060648
  10. A. Elgart, L. Erdős, B. Schlein, and H.-T. Yau. Nonlinear Hartree equation as the mean field limit of weakly coupled fermions. J. Math. Pures Appl. (9)83 (2004), no. 10, 1241–1273. Zbl1059.81190MR2092307
  11. L. Erdős, B. Schlein, and H.-T. Yau. Derivation of the cubic non-linear Schrödinger equation from quantum dynamics of many-body systems. Inv. Math.167 (2006), 515–614. Zbl1123.35066MR2276262
  12. J. Fröhlich and A. Knowles. A microscopic derivation of the time-dependent Hartree-Fock equation with Coulomb two-body interaction. J. Stat. Phys.145 (2011), no. 1, 23–50. Zbl1269.82042MR2841931
  13. J. Fröhlich and E. Lenzmann. Dynamical collapse of white dwarfs in Hartree- and Hartree-Fock theory. Comm. Math. Phys.274 (2007), 737–750. Zbl1130.85004MR2328910
  14. G. M. Graf and J. P. Solovej. A correlation estimate with applications to quantum systems with Coulomb interactions. Rev. Math. Phys.6 (1994), 977–997. Zbl0843.47041MR1301362
  15. C. Hainzl and B. Schlein. Stellar collapse in the time-dependent Hartree-Fock approximation. Comm. Math. Phys.287 (2009), 705–717. Zbl1175.85002MR2481756
  16. O. E. Lanford III. The evolution of large classical system. Dynamical Systems, theory and applications. Lecture Notes in Physics38, 1–111 (1975). Zbl0329.70011MR479206
  17. E. H. Lieb, R. Seiringer, J. .P. Solovej and J. Yngvason. The mathematics of the Bose gas and its condensation. Oberwolfach seminars34, Birkhäuser (2005). Zbl1104.82012MR2143817
  18. E. H. Lieb and B. Simon. The Thomas-Fermi theory of atoms, molecules and solids. Adv. Math.23, 22–116 (1977). Zbl0938.81568MR428944
  19. P. L. Lions and T. Paul. Sur les mesures de Wigner. Revista matemática iberoamericana9, 553–618 (1993). Zbl0801.35117MR1251718
  20. P. A. Markowich and N. J. Mauser. The classical limit of a self-consistent quantum-Vlasov equation in 3D. Mathematical Models and Methods in Applied Sciences3, 109 (1993). Zbl0772.35061MR1203274
  21. H. Narnhofer and G. L. Sewell. Vlasov hydrodynamics of a quantum mechanical model. Comm. Math. Phys.79 (1981), no. 1, 9–24. MR609224
  22. S. Petrat and P. Pickl. A New Method and a New Scaling For Deriving Fermionic Mean-field Dynamics. http://arxiv.org/abs/1409.0480 
  23. J. P. Solovej. Many Body Quantum Mechanics. Lecture Notes. Summer 2007. Available at http://www.mathematik.uni-muenchen.de/~sorensen/Lehre/SoSe2013/MQM2/skript.pdf 
  24. H. Spohn. On the Vlasov hierarchy, Math. Methods Appl. Sci.3 (1981), no. 4, 445–455. Zbl0492.35067MR657065

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