Quantitative Derivation of the Gross-Pitaevskii Equation

TL;DR

This paper rigorously derives the Gross-Pitaevskii equation from many-body quantum dynamics, providing bounds on convergence rates and emphasizing the role of Bogoliubov transformations in modeling Bose-Einstein condensates.

Contribution

It offers a quantitative derivation of the Gross-Pitaevskii equation from first principles, including explicit convergence bounds and the construction of initial data using Bogoliubov transformations.

Findings

Convergence of many-body dynamics to the Gross-Pitaevskii equation with rate N^{-1/2}

Initial data constructed via Bogoliubov transformation preserves microscopic correlations

Bound on the rate of convergence of the reduced density matrix

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 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 which vanishes as N^{-1/2} in the limit of large N.