Effective Evolution Equations from Quantum Dynamics

Niels Benedikter, Marcello Porta, Benjamin Schlein (Autoren)

Buch | Softcover

VII, 91 Seiten

2015 | 1st ed. 2016
Springer International Publishing (Verlag)
978-3-319-24896-7 (ISBN)

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These notes investigate the time evolution of quantum systems, and in particular the rigorous derivation of effective equations approximating the many-body Schrödinger dynamics in certain physically interesting regimes. The focus is primarily on the derivation of time-dependent effective theories (non-equilibrium question) approximating many-body quantum dynamics. The book is divided into seven sections, the first of which briefly reviews the main properties of many-body quantum systems and their time evolution. Section 2 introduces the mean-field regime for bosonic systems and explains how the many-body dynamics can be approximated in this limit using the Hartree equation. Section 3 presents a method, based on the use of coherent states, for rigorously proving the convergence towards the Hartree dynamics, while the fluctuations around the Hartree equation are considered in Section 4. Section 5 focuses on a discussion of a more subtle regime, in which the many-body evolution can be approximated by means of the nonlinear Gross-Pitaevskii equation. Section 6 addresses fermionic systems (characterized by antisymmetric wave functions); here, the fermionic mean-field regime is naturally linked with a semiclassical regime, and it is proven that the evolution of approximate Slater determinants can be approximated using the nonlinear Hartree-Fock equation. In closing, Section 7 reexamines the same fermionic mean-field regime, but with a focus on mixed quasi-free initial data approximating thermal states at positive temperature.

Introduction.- Mean-Field Regime for Bosonic Systems.- Coherent States Approach.-Fluctuations Around Hartree Dynamics.- The Gross-Pitaevskii Regime.- Mean-Field regime for Fermionic Systems.- Dynamics of Fermionic Quasi-Free Mixed States.- The Role of Correlations in the Gross-Pitaevskii Energy.

Erscheinungsdatum	08.10.2016
Reihe/Serie	SpringerBriefs in Mathematical Physics
Zusatzinfo	VII, 91 p.
Verlagsort	Cham
Sprache	englisch
Maße	155 x 235 mm
Themenwelt	Naturwissenschaften ► Physik / Astronomie ► Quantenphysik
Themenwelt	Naturwissenschaften ► Physik / Astronomie ► Theoretische Physik
Schlagworte	Bosonic Mean-field Regime • Coherent States Approach • Fermionic Mean-field Regime • Gross-Pitaevskii Equation • Hartree Equation • Hartree-Fock Equation • Many-body Quantum Dynamics • Many-body quantum systems • Many-body Schrödinger Dynamics • Mathematical Physics • Physics and Astronomy • Quantum Physics • Schrödinger Dynamics • Slater Determinants
ISBN-10	3-319-24896-0 / 3319248960
ISBN-13	978-3-319-24896-7 / 9783319248967
Zustand	Neuware