Abstract
We consider the evolution of quasi-free states describing N fermions in the mean field limit, as governed by the nonlinear Hartree equation. In the limit of large N, we study the convergence towards the classical Vlasov equation. For a class of regular interaction potentials, we establish precise bounds on the 0rate of convergence.
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Amour L., Khodja M., Nourrigat J.: The semiclassical limit of the time dependent Hartree–Fock equation: the Weyl symbol of the solution. Anal. PDE 6(7), 1649–1674 (2013)
Amour L., Khodja M., Nourrigat J.: The classical limit of the Heisenberg and time dependent Hartree–Fock equations: the Wick symbol of the solution. Math. Res. Lett. 20(1), 119–139 (2013)
Athanassoulis A., Paul T., Pezzotti F., Pulvirenti M.: Strong semiclassical approximation of Wigner functions for the Hartree dynamics. Rend. Lincei Mat. Appl. 22, 525–552 (2011)
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 (preprint). arXiv:1403.1488
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)
Benedikter, N., Jaksic, V., Porta, M., Saffirio, C., Schlein, B.: Mean-field evolution of fermionic mixed states Comm. Pure Appl. Math. doi:10.1062/cpa.21538
Benedikter N., Porta M., Schlein B.: Mean-field evolution of fermionic systems. Commun. Math. Phys. 331, 1087–1131 (2014)
Dobrushin R.L.: Vlasov equations. Funct. Anal. Appl. 13(2), 115–123 (1979)
Elgart A., Erdos 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)
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)
Gasser I., Illner R., Markowich P.A., Schmeiser C.: Semiclassical, t → ∞ asymptotics and dispersive effects for HF systems. Math. Model. Numer. Anal. 32, 699–713 (1998)
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)
Lions P.-L., Paul T.: Sur les mesures de Wigner. Rev. Mat. Iberoam. 9, 553–618 (1993)
Markowich P.A., Mauser N.J.: The classical limit of a self-consistent quantum Vlasov equation. Math. Models Methods Appl. Sci. 3(1), 109–124 (1993)
Narnhofer H., Sewell G.L.: Vlasov hydrodynamics of a quantum mechanical model. Commun. Math. Phys. 79(1), 9–24 (1981)
Prodan E., Nordlander P.: Hartree approximation I: The fixed point approach. J. Math. Phys. 42, 3390–3406 (2001)
Pezzotti F., Pulvirenti M.: Mean-field limit and semiclassical expansion of a quantum particle system. Ann. Henri Poincaré 10(1), 145–187 (2009)
Petrat S., Pickl P.: A new method and a new scaling for deriving fermionic mean-field dynamics. arXiv:1409.0480
Prodan E.: Symmetry breaking in the self-consistent Kohn–Sham equations. J. Phys. A Math. Gen. 38, 5647–5657 (2005)
Solovej J.P.; Many body quantum mechanics. In: Lecture Notes, Summer 2007. https://www.mathematik.uni-muenchen.de/~lerdos/WS08/QM/solovejnotes.pdf
Spohn H.: On the Vlasov hierarchy. Math. Methods Appl. Sci. 3(4), 445–455 (1981)
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Benedikter, N., Porta, M., Saffirio, C. et al. From the Hartree Dynamics to the Vlasov Equation. Arch Rational Mech Anal 221, 273–334 (2016). https://doi.org/10.1007/s00205-015-0961-z
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DOI: https://doi.org/10.1007/s00205-015-0961-z