×

zbMATH — the first resource for mathematics

Mean field dynamics of fermions and the time-dependent Hartree-Fock equation. (English) Zbl 1029.82022
Summary: The time-dependent Hartree-Fock equations are derived from the \(N\)-body linear Schrödinger equation with the mean-field scaling in the limit \(N\to +{\infty}\) and for initial data that are close to Slater determinants. Only the case of bounded, symmetric binary interaction potentials is treated in this work. We prove that, as \(N\to +{\infty}\), the first partial trace of the \(N\)-body density operator approaches the solution of the time-dependent Hartree-Fock equations (in operator form) in the sense of the trace norm.

MSC:
82C10 Quantum dynamics and nonequilibrium statistical mechanics (general)
82C22 Interacting particle systems in time-dependent statistical mechanics
35Q55 NLS equations (nonlinear Schrödinger equations)
47N50 Applications of operator theory in the physical sciences
PDF BibTeX XML Cite
Full Text: DOI arXiv
References:
[1] Alicki, R.; Messer, J., Nonlinear quantum dynamical semigroups for many-body open systems, J. statist. phys., 32, 2, 299-312, (1983) · Zbl 0584.35054
[2] Bardos, C.; Golse, F.; Mauser, N.J., Weak coupling limit of the N-particle Schrödinger equation, Math. anal. appl., 7, 2, 275-293, (2000) · Zbl 1003.81027
[3] Bardos, C.; Erdös, L.; Golse, F.; Mauser, N.J.; Yau, H.-T., Derivation of the schrödinger – poisson equation from the quantum N-particle Coulomb problem, C. R. acad. sci., (2002)
[4] Boltzmann, L., Lectures on gas theory, (1995), Dover New York
[5] Bove, A.; Da Prato, G.; Fano, G., An existence proof for the hartree – fock time-dependent problem with bounded two-body interaction, Comm. math. phys., 37, 183-191, (1974) · Zbl 0303.34046
[6] Cances, E.; Le Bris, C., On the time-dependent hartree – fock equations coupled with a classical nuclear dynamics, Math. models methods appl. sci., 9, 7, 963-990, (1999) · Zbl 1011.81087
[7] Duffield, N.G.; Werner, R.F., Local dynamics of Mean-field quantum systems, Helv. phys. acta, 65, 1016-1054, (1992) · Zbl 0771.46035
[8] L. Erdös, H.-T. Yau, Derivation of the nonlinear Schrödinger equation with Coulomb potential, Preprint, 2002
[9] A.D. Gottlieb, Markov transitions and the propagation of chaos, PhD Thesis, Lawrence Berkeley National Laboratory Report, LBNL-42839, 1998
[10] Grünbaum, F.A., Propagation of chaos for the Boltzmann equation, Arch. rational mech. anal., 42, 323-345, (1971) · Zbl 0236.45011
[11] Kac, M., Foundations of kinetic theory, () · Zbl 0072.42802
[12] Kac, M., Probability and related topics in physical sciences, (1976), Amer. Math. Society Providence, RI
[13] McKean, H.P., A class of Markov processes associated with nonlinear parabolic equations, Proc. nat. acad. sci., 56, 1907-1911, (1966) · Zbl 0149.13501
[14] Méléard, S., Asymptotic behavior of some interacting particle systems; mckean – vlasov and Boltzmann models, Lecture notes in math., 1627, (1995), Springer-Verlag Berlin
[15] Spohn, H., Kinetic equations from Hamiltonian dynamics, Rev. mod. phys., 53, 600-640, (1980), see Theorem 5.7
[16] Sznitman, A., Équations de type de Boltzmann, spatialement homogènes, Z. wahrscheinlichkeitsth., 66, 559-592, (1984) · Zbl 0553.60069
[17] Sznitman, A., Topics in propagation of chaos, Lecture notes in math., 1464, (1991), Springer-Verlag Berlin
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.