Mean-field dynamics of fermions with relativistic dispersion

Benedikter, Niels; Porta, Marcello; Schlein, Benjamin

doi:10.1063/1.4863349

Cited by 47 publications

(62 citation statements)

References 12 publications

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“…Actually, it turns out that Hartree-Fock theory provides more than an upper bound for the ground state energy: the method developed in [2,3,33] for the jellium model can also be applied to show that in the present mean-field scaling the Hartree-Fock minimum agrees with the many-body ground state energy up to an error of size o (1) for N → ∞. Moreover, by projection of the time-dependent Schrödinger equation onto the manifold of quasi-free states one obtains the time-dependent Hartree-Fock equation [10], which was proven to effectively approximate the many-body evolution of mean-field fermionic systems [5,7,6,8,58,59]. For N non-interacting particles on the torus, the ground state is given by the Slater determinant constructed from plane waves…”

Section: Introductionmentioning

confidence: 70%

Optimal Upper Bound for the Correlation Energy of a Fermi Gas in the Mean-Field Regime

Benedikter

Nam

Porta

et al. 2019

Commun. Math. Phys.

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158

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While Hartree-Fock theory is well established as a fundamental approximation for interacting fermions, it has been unclear how to describe corrections to it due to manybody correlations. In this paper we start from the Hartree-Fock state given by plane waves and introduce collective particle-hole pair excitations. These pairs can be approximately described by a bosonic quadratic Hamiltonian. We use Bogoliubov theory to construct a trial state yielding a rigorous Gell-Mann-Brueckner-type upper bound to the ground state energy. Our result justifies the random-phase approximation in the mean-field scaling regime, for repulsive, regular interaction potentials.

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Section: Introductionmentioning

confidence: 70%

Optimal Upper Bound for the Correlation Energy of a Fermi Gas in the Mean-Field Regime

Benedikter

Nam

Porta

et al. 2019

Commun. Math. Phys.

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158

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“…Hence, each frequency must pair up with at least one of the other frequencies or else the contribution equals zero. This observation reduces the original sum one is considering in (16). The claim then follows by using Hölder's inequality.…”

Section: Ideas and Techniques Used In The Proofsmentioning

confidence: 90%

“…The derivation of a hierarchy similar to (1) coming from the limit of N -body Schrödinger dynamics was obtained in [1,2,11,[15][16][17]45,48,[51][52][53][54][55][56][57]74,[84][85][86][87]124,162] in various different contexts. The first rate of convergence result was obtained by Rodnianski and Schlein [145] and subsequent rate of convergence results have been obtained in [6,17,38,49,50,75,85,[93][94][95]110,115,121,124,137,138].…”

Section: Previously Known Resultsmentioning

confidence: 99%

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Randomization and the Gross–Pitaevskii Hierarchy

Sohinger

Staffilani

2015

Arch Rational Mech Anal

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We study the Gross-Pitaevskii hierarchy on the spatial domain T 3 . By using an appropriate randomization of the Fourier coefficients in the collision operator, we prove an averaged form of the main estimate which is used in order to contract the Duhamel terms that occur in the study of the hierarchy. In the averaged estimate, we do not need to integrate in the time variable. An averaged spacetime estimate for this range of regularity exponents then follows as a direct corollary. The range of regularity exponents that we obtain is α > 3 4 . It was shown in our previous joint work with Gressman (J Funct Anal 266(7): 2014) that the range α > 1 is sharp in the corresponding deterministic spacetime estimate. This is in contrast to the non-periodic setting, which was studied by Klainerman and Machedon (Commun Math Phys 279(1): [169][170][171][172][173][174][175][176][177][178][179][180][181][182][183][184][185] 2008), where the spacetime estimate is known to hold whenever α ≥ 1. The goal of our paper is to extend the range of α in this class of estimates in a probabilistic sense. We use the new estimate and the ideas from its proof in order to study randomized forms of the Gross-Pitaevskii hierarchy. More precisely, we consider hierarchies similar to the Gross-Pitaevskii hierarchy, but in which the collision operator has been randomized. For these hierarchies, we show convergence to zero in low regularity Sobolev spaces of Duhamel expansions of fixed deterministic density matrices. We believe that the study of the randomized collision operators could be the first step in the understanding of a nonlinear form of randomization.

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“…(iii) The result has been extended in [6] to the dynamics of pseudorelativistic fermions. That is, the Laplacian is replaced by √ −ε 2 ∆ + m 2 , with m = O(1).…”

Section: Viii-7mentioning

confidence: 99%