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 that vanishes as N-1/2 in the limit of large N.
Quantitative Derivation of the Gross-Pitaevskii Equation / N. Benedikter, G. de Oliveira, B. Schlein. - In: COMMUNICATIONS ON PURE AND APPLIED MATHEMATICS. - ISSN 0010-3640. - 68:8(2015), pp. 1399-1482. [10.1002/cpa.21542]
Quantitative Derivation of the Gross-Pitaevskii Equation
N. Benedikter;
2015
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 that vanishes as N-1/2 in the limit of large N.File | Dimensione | Formato | |
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