Abstract
We introduce a new method for deriving the time-dependent Hartree or Hartree-Fock equations as an effective mean-field dynamics from the microscopic Schrödinger equation for fermionic many-particle systems in quantum mechanics. The method is an adaption of the method used in Pickl (Lett. Math. Phys. 97 (2) 151–164 2011) for bosonic systems to fermionic systems. It is based on a Gronwall type estimate for a suitable measure of distance between the microscopic solution and an antisymmetrized product state. We use this method to treat a new mean-field limit for fermions with long-range interactions in a large volume. Some of our results hold for singular attractive or repulsive interactions. We can also treat Coulomb interaction assuming either a mild singularity cutoff or certain regularity conditions on the solutions to the Hartree(-Fock) equations. In the considered limit, the kinetic and interaction energy are of the same order, while the average force is subleading. For some interactions, we prove that the Hartree(-Fock) dynamics is a more accurate approximation than a simpler dynamics that one would expect from the subleading force. With our method we also treat the mean-field limit coupled to a semiclassical limit, which was discussed in the literature before, and we recover some of the previous results. All results hold for initial data close (but not necessarily equal) to antisymmetrized product states and we always provide explicit rates of convergence.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
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)
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)
Anapolitanos, I.: Rate of convergence towards the Hartree-von Neumann limit in the mean-field regime. Lett. Math. Phys. 98(1), 1–31 (2011)
Athanassoulis, A., Paul, T., Pezzotti, F., Pulvirenti, M.: Strong semiclassical approximation of Wigner functions for the Hartree dynamics. Rend Lincei - Mat. Appl. 22(4), 525–552 (2011)
Bach, V.: Accuracy of mean field approximations for atoms and molecules. Commun. Math. Phys. 155(2), 295–310 (1993)
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), 1–30 (2016)
Bardos, C., Ducomet, B., Golse, F., Gottlieb, A.D., Mauser, N.J.: The TDHF approximation for Hamiltonians with m-particle interaction potentials. Commun. Math. Sci. 5, 1–9 (2007)
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. 82 (6), 665–683 (2003)
Bardos, C., Golse, F., Gottlieb, A.D., Mauser, N.J.: Accuracy of the time-dependent Hartree-Fock approximation for uncorrelated initial states. J. Stat. Phys. 115(3–4), 1037–1055 (2004)
Benedikter, N., de Oliveira, G., Schlein, B.: Quantitative derivation of the Gross-Pitaevskii equation. Commun. Pure Appl. Math. 68(8), 1399–1482 (2015)
Benedikter, N., Jakšić, V., Porta, M., Saffirio, C., Schlein, B.: Mean-field evolution of fermionic mixed states. Commun. Pure Appl. Math. (2015). In press
Benedikter, N., Porta, M., Saffirio, C., Schlein, B.: From the Hartree dynamics to the Vlasov equation. Arch. Ration. Mech. Anal. (2016). In press
Benedikter, N., Porta, M., Schlein, B.: Mean-field dynamics of fermions with relativistic dispersion. J. Math. Phys. 55(2) (2014)
Benedikter, N., Porta, M., Schlein, B.: Mean-field evolution of fermionic systems. Commun. Math. Phys. 331(3), 1087–1131 (2014)
Bove, A., Da Prato, G., Fano, G.: An existence proof for the Hartree-Fock time-dependent problem with bounded two-body interaction. Commun. Math. Phys. 37(3), 183–191 (1974)
Bove, A., Da Prato, G., Fano, G.: On the Hartree-Fock time-dependent problem. Commun. Math. Phys. 49(1), 25–33 (1976)
Braun, W., Hepp, K.: The Vlasov dynamics and its fluctuations in the 1/N limit of interacting classical particles. Commun. Math. Phys. 56(2), 101–113 (1977)
Chadam, J.M.: The time-dependent Hartree-Fock equations with Coulomb two-body interaction. Commun. Math. Phys. 46(2), 99–104 (1976)
Chadam, J.M., Glassey, R.T.: Global existence of solutions to the Cauchy problem for time-dependent Hartree equations. J. Math. Phys. 16(5), 1122–1130 (1975)
Deckert, D.-A., Fröhlich, J., Pickl, P., Pizzo, A.: Dynamics of sound waves in an interacting Bose gas. Preprint (2014). arXiv:http://arXiv.org/abs/1406.1590
Deckert, D.-A., Fröhlich, J., Pickl, P., Pizzo, A.: Effective dynamics of a tracer particle interacting with an ideal Bose gas. Commun. Math. Phys. 328(2), 597–624 (2014)
Elgart, A., Erdös, L., Schlein, B., Yau, H.-T.: Nonlinear Hartree equation as the mean field limit of weakly coupled fermions. J. Math. Pures Appl. 83(10), 1241–1273 (2004)
Erdös, L., Schlein, B., Yau, H.-T.: Derivation of the Gross-Pitaevskii hierarchy for the dynamics of Bose-Einstein condensate. Commun. Pure Appl. Math. 59(12), 1659–1741 (2006)
Erdős, L., Schlein, B., Yau, H.-T.: Derivation of the cubic non-linear Schrödinger equation from quantum dynamics of many-body systems. Invent. Math. 167(3), 515–614 (2007)
Erdős, L., Schlein, B., Yau, H.-T.: Rigorous derivation of the Gross-Pitaevskii equation. Phys. Rev. Lett. 98(4) (2007)
Erdős, L., Schlein, B., Yau, H.-T.: Rigorous derivation of the Gross-Pitaevskii equation with a large interaction potential. J. Am. Math. Soc. 22(4), 1099–1156 (2009)
Erdős, L., Schlein, B., Yau, H.-T.: Derivation of the Gross-Pitaevskii equation for the dynamics of Bose-Einstein condensate. Ann. Math. 172(1), 291–370 (2010)
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)
Frank, R., Hoffmann-Ostenhof, T., Laptev, A., Solovej, J. P.: Hardy inequalities for large fermionic systems. work in progress (2006)
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)
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)
Fröhlich, J., Lenzmann, E.: Dynamical collapse of white dwarfs in Hartree- and Hartree-Fock theory. Commun. Math. Phys. 274(3), 737–750 (2007)
Gasser, I., Illner, R., Markowich, P.A., Schmeiser, C.: Semiclassical, t → ∞ asymptotics and dispersive effects for Hartree-Fock systems. Math. Modell. Numer. Anal. 32(6), 699–713 (1998)
Graf, G.M., Solovej, J.P.: A correlation estimate with applications to quantum systems with Coulomb interaction. Rev. Math. Phys. 6(5a), 977–997 (1994)
Hainzl, C., Lenzmann, E., Lewin, M., Schlein, B.: On blowup for time-dependent generalized Hartree-Fock equations. Ann. Henri Poincaré 11(6), 1023–1052 (2010)
Hainzl, C., Schlein, B.: Stellar collapse in the time dependent Hartree-Fock approximation. Commun. Math. Phys. 287(2), 705–717 (2009)
Hepp, K.: The classical limit for quantum mechanical correlation functions. Commun. Math. Phys. 35(4), 265–277 (1974)
Hoffmann-Ostenhof, M., Hoffmann-Ostenhof, T., Laptev, A., Tidblom, J.: Many particle Hardy inequalities. J. London Math. Soc. 77, 99–114 (2008)
Knowles, A., Pickl, P. : Mean-field dynamics: Singular potentials and rate of convergence. Commun. Math. Phys. 298(1), 101–138 (2010)
Lanford, O.E.: Time evolution of large classical systems. In: Moser, J. (ed.) Dynamical systems, theory and applications, volume 38 of Lecture Notes in Physics, pp. 1–111. Springer (1975)
Lieb, E.: Thomas-fermi and related theories of atoms and molecules. Rev. Mod. Phys. 53(4), 603–641 (1981)
Lieb, E., Loss, M.: Analysis. American Mathematical Society, second edition (2001)
Lieb, E., Seiringer, R.: The Stability of Matter in Quantum Mechanics. Cambridge University Press (2010)
Lieb, E., Thirring, W.E.: Bound for the kinetic energy of fermions which proves the stability of matter. Phys. Rev Lett. 35(11), 687–689 (1975)
Lions, P.L., Paul, T.: Sur les mesures de Wigner. Mat. Rev. Iberoamericana 9(3), 553–618 (1993)
Markowich, P.A., Mauser, N.J.: The classical limit of a self-consistent quantum-Vlasov equation in 3D. 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)
Petrat, S.: Derivation of Mean-field Dynamics for Fermions. PhD thesis, Ludwig-Maximilians-Universität München (2014)
Pezzotti, F., Pulvirenti, M.: Mean-field limit and semiclassical expansion of a quantum particle system. Ann. Henri Poincaré 10(1), 145–187 (2009)
Pickl, P.: Derivation of the time dependent Gross-Pitaevskii equation without positivity condition on the interaction. J. Stat. Phys. 140(1), 76–89 (2010)
Pickl, P.: A simple derivation of mean field limits for quantum systems. Lett. Math. Phys. 97(2), 151–164 (2011)
Pickl, P.: Derivation of the time dependent Gross-Pitaevskii equation with external fields. Rev. Math. Phys. 27(1) (2015)
Reed, M., Simon, B.: Methods of Modern Mathematical Physics. I: Functional Analysis, first edition. Academic Press Inc. (1980)
Rodnianski, I., Schlein, B.: Quantum fluctuations and rate of convergence towards mean field dynamics. Commun. Math. Phys. 291(1), 31–61 (2009)
M. Rumin.: Balanced distribution-energy inequalities and related entropy bounds. Duke Math. J. 160(3), 567–597 (2011)
Spohn, H.: Kinetic equations from Hamiltonian dynamics: Markovian limits. Rev. Mod. Phys. 53(3), 569–615 (1980)
Spohn, H.: On the Vlasov hierarchy. Math. Methods Appl. Sci. 3(1), 445–455 (1981)
Spohn, H.: Large Scale Dynamics of Interacting Particles, first edition. Springer (1991)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
About this article
Cite this article
Petrat, S., Pickl, P. A New Method and a New Scaling for Deriving Fermionic Mean-Field Dynamics. Math Phys Anal Geom 19, 3 (2016). https://doi.org/10.1007/s11040-016-9204-2
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s11040-016-9204-2