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arXiv:1012.4734

Effective evolution equations from many body quantum dynamics

Benjamin Schlein

Published Dec 21, 2010• 0 citations
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Abstract

In these notes we review some recent results concerning the derivation of effective equations from first principle quantum dynamics. In particular, we discuss the derivation of the semi-relativistic Hartree equation for the evolution of boson stars, and the derivation of the Gross-Pitaevskii equation for the dynamics of Bose-Einstein condensates.

DOI
10.48550/arxiv.1012.4734

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In these notes we review some recent results concerning the derivation of effective equations from first principle quantum dynamics. In particular, we discuss the derivation of the semi-relativistic Hartree equation for the evolution of boson stars, and the derivation of the Gross-Pitaevskii equation for the dynamics of Bose-Einstein condensates.
In these notes, we review some recent mathematical results concerning the derivation of effective evolution equations from many body quantum mechanics. In particular, we discuss the emergence of the Hartree equation in the so-called mean field regime (for example, for systems of gravitating bosons), and we show that the Gross-Pitaevskii equation approximates the dynamics of initially trapped Bose-Einstein condensates. We explain how effective evolution equations can be derived, on the one hand, by analyzing the so called BBGKY hierarchy, describing the time-evolution of reduced density matrices, and, on the other hand, by studying the dynamics of coherent initial states in a Fock-space representation of the many body system.
In these notes, based on a mini-course held at the summer school "Methods and Models of Kinetic Theory" that took place in Porto Ercole in June 2016, we review some of the recent developments in the derivation of effective evolution equations starting from many-body quantum mechanics. We discuss the derivation of the Hartree equation in the bosonic mean-field limit, of the Gross-Pitaevskii equation describing the dynamics of initially trapped Bose-Einstein condensates and of the Hartree-Fock equation for fermions in a joint mean-field and semiclassical limit.
In experiments, Bose-Einstein condensates are prepared by cooling a dilute Bose gas in a trap. After the phase transition has been reached, the trap is switched off and the evolution of the condensate observed. The evolution is macroscopically described by the Gross-Pitaevskii equation. On the microscopic level, the dynamics of Bose gases are described by the NNN-body Schrödinger equation. We review our article [BdS12] in which we construct a class of initial data in Fock space which are energetically close to the ground state and prove that their evolution approximately follows the Gross-Pitaevskii equation. The key idea is to model two-particle correlations with a Bogoliubov transformation.
We study the time evolution of Bose--Einstein condensates with three-body interactions in the Gross--Pitaevskii regime. We show that Bose--Einstein condensation is preserved under many-body evolution and that the condensate wavefunction evolves according to the quintic Gross--Pitaevskii equation in R3\mathbb{R}^3R3, which is energy critical. In particular, we show that the effective coupling constant is universal and depends only on a three-body scattering hypervolume.