Mean field dynamics of fermions and the time-dependent Hartree–Fock equation

Bardos, Claude; Golse, François; Gottlieb, Alex D.; Mauser, Norbert J.

doi:10.1016/s0021-7824(03)00023-0

Cited by 110 publications

(177 citation statements)

References 18 publications

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“…In fact, the result in [3] shows that the same result holds verbatim if D N (t) is the solution of the N -body von Neumann equation (II.6) with initial data D N (0) having Slater closure, while F (t) is the solution to the TDHF equation (III.19) with initial data D N :1 (0).…”

Section: Ix-38mentioning

confidence: 75%

“…The TDHF approximation for bounded 2-body interaction is summarized in the following Theorem, taken from [3]: …”

Section: Iii3 Convergence Of the N -Body Problem To The Tdhf Equationmentioning

confidence: 99%

“…The interested reader is referred to [3] for a complete proof; here we just sketch the main ideas in it.…”

Section: Iii4 Sketch Of the Proof Of The Tdhf Approximationmentioning

confidence: 99%

“…The proof is based on the ideas in [3] and [15]. It requires more than the uniqueness estimate in [15], namely an estimate of the difference between both hierarchies (i.e.…”

Section: Ix-43mentioning

confidence: 99%

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The mean-field limit for the dynamics of large particle systems

Golse¹

2003

Journées Équations Aux Dérivées Partielles

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101

168

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This short course explains how the usual mean-field evolution PDEs in Statistical Physics -such as the Vlasov-Poisson, Schrödinger-Poisson or time-dependent Hartree-Fock equations -are rigorously derived from first principles, i.e. from the fundamental microscopic models that govern the evolution of large, interacting particle systems. A review of physical modelsThe subject matter of these lectures is the relation between "exact" microscopic models that govern the evolution of large particle systems and a certain type of approximate models known in Statistical Mechanics as "mean-field equations". This notion of mean-field equation is best understood by getting acquainted with the most famous examples of such equations, described below.

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Section: Ix-38mentioning

confidence: 75%

“…The TDHF approximation for bounded 2-body interaction is summarized in the following Theorem, taken from [3]: …”

Section: Iii3 Convergence Of the N -Body Problem To The Tdhf Equationmentioning

confidence: 99%

“…The interested reader is referred to [3] for a complete proof; here we just sketch the main ideas in it.…”

Section: Iii4 Sketch Of the Proof Of The Tdhf Approximationmentioning

confidence: 99%

“…The proof is based on the ideas in [3] and [15]. It requires more than the uniqueness estimate in [15], namely an estimate of the difference between both hierarchies (i.e.…”

Section: Ix-43mentioning

confidence: 99%

See 2 more Smart Citations

The mean-field limit for the dynamics of large particle systems

Golse¹

2003

Journées Équations Aux Dérivées Partielles

Self Cite

101

168

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“…However, the attempts to incorporate the resulting "exchange interaction" as a local term in the time-dependent S-P model, are heuristic (cf [28]) up to now. Steps in this direction are the recent progress on the derivation of the time dependent Hartree-Fock (HF) equation [4], [5] and the derivation of the "Xα" potential as a local approximation of the HF exchange potential [10], [11].…”

Section: Introductionmentioning

confidence: 99%

Semi-classical limits of Schrödinger-Poisson systems via Wigner transforms

Mauser¹

2002

Journées Équations Aux Dérivées Partielles

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We deal with classical and "semiclassical limits" , i.e. vanishing Planck constant → 0, eventually combined with a homogenization limit of a crystal lattice, of a class of "weakly nonlinear" NLS. The Schrödinger-Poisson (S-P) system for the wave functions {ψ j (t, x)} is transformed to the WignerPoisson (W-P) equation for a "phase space function" f (t, x, ξ), the Wigner function. The weak limit of f (t, x, ξ), as tends to 0, is called the "Wigner measure" f (t, x, ξ) (also called "semiclassical measure" by P. Gérard).The mathematically rigorous classical limit from S-P to the Vlasov-Poisson (V-P) system has been solved first in 1993 by P.L. Lions and T. Paul in [21] and, independently, by P.A. Markowich and N.J. Mauser in [23]. There the case of the so called "completely mixed state", i.e. j = 1, 2, . . . , ∞, was considered where strong additional assumptions can be posed on the initial data.For the so called "pure state" case where only one (or a finite number) of wave functions {ψ j (t, x)} is considered, recently P. Zhang, Y. Zheng and N.J. Mauser [33] have given the limit from S-P to V-P in one space dimension for a very weak class of measure valued solutions of V-P that are not unique.For the setting in a crystal, as it occurs in semiconductor modeling, we consider Schrödinger equations with an additional periodic potential. This allows for the use of the concept of "energy bands", Bloch decomposition of L 2 etc. On the level of the Wigner transform the Wigner function f (t, x, ξ) is replaced by the "Wigner series" f (t, x, k), where the "kinetic variable" k lives on the torus ("Brioullin zone") instead of the whole space.Recently P. Bechouche, N.J. Mauser and F. Poupaud [7] have given the rigorous "semiclassical" limit from S-P in a crystal to the "semiclassical equations", i.e. the "semiconductor V-P system", with the assumption of the initial data to be concentrated in isolated bands.Support by the Austrian START project "Nonlinear Schrödinger and quantum Boltzmann equations" is acknowledged. MSC 2000 : 35L65, 35Q55, 81Q05, 81Q20.

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Mean‐Field Evolution of Fermionic Mixed States

Benedikter

Jakšić

Porta

et al. 2015

Comm Pure Appl Math

130

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In this paper we study the dynamics of fermionic mixed states in the mean-field regime. We consider initial states that are close to quasi-free states and prove that, under suitable assumptions on the initial data and on the many-body interaction, the quantum evolution of such initial data is well approximated by a suitable 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 result holds for all times and gives effective estimates on the rate of convergence of the many-body dynamics towards the Hartree-Fock evolution.

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Mean field dynamics of fermions and the time-dependent Hartree–Fock equation

Cited by 110 publications

References 18 publications

The mean-field limit for the dynamics of large particle systems

The mean-field limit for the dynamics of large particle systems

Semi-classical limits of Schrödinger-Poisson systems via Wigner transforms

Mean‐Field Evolution of Fermionic Mixed States

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