# Quantitative erivation of the Gross-Pitaevskii equation

Benedikter, Niels; de Oliveira, Gustavo; Schlein, Benjamin (2015). Quantitative erivation of the Gross-Pitaevskii equation. Communications on Pure and Applied Mathematics, 68(8):1399-1482.

## Abstract

Starting from first-principle many-body quantum dynamics, we show that the dynamics of Bose-Einstein condensates can be approximated by the time-dependent nonlinear Gross-Pitaevskii equation, giving a bound on the rate of the convergence. Initial data are constructed on the bosonic Fock space applying an appropriate Bogoliubov transformation on a coherent state with expected number of particles $\mathit{N}$. The Bogoliubov transformation plays a crucial role; it produces the correct microscopic correlations among the particles. Our analysis shows that, on the level of the one-particle reduced density, the form of the initial data is preserved by the many-body evolution, up to a small error that vanishes as $\mathit{N}^{-1/2}$ in the limit of large $\mathit{N}$.

## Abstract

Starting from first-principle many-body quantum dynamics, we show that the dynamics of Bose-Einstein condensates can be approximated by the time-dependent nonlinear Gross-Pitaevskii equation, giving a bound on the rate of the convergence. Initial data are constructed on the bosonic Fock space applying an appropriate Bogoliubov transformation on a coherent state with expected number of particles $\mathit{N}$. The Bogoliubov transformation plays a crucial role; it produces the correct microscopic correlations among the particles. Our analysis shows that, on the level of the one-particle reduced density, the form of the initial data is preserved by the many-body evolution, up to a small error that vanishes as $\mathit{N}^{-1/2}$ in the limit of large $\mathit{N}$.

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15 citations in Web of Science®
13 citations in Scopus®

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Item Type: Journal Article, refereed, original work 07 Faculty of Science > Institute of Mathematics 510 Mathematics English August 2015 03 Feb 2016 08:20 05 Apr 2016 19:51 Wiley-Blackwell Publishing, Inc. 0010-3640 https://doi.org/10.1002/cpa.21542

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