Effective Evolution Equations from Quantum Dynamics
Authors: Benedikter, Niels, Porta, Marcello, Schlein, Benjamin
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- About this book
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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.
- Table of contents (7 chapters)
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Introduction
Pages 1-6
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Mean-Field Regime for Bosonic Systems
Pages 7-16
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Coherent States Approach
Pages 17-29
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Fluctuations Around Hartree Dynamics
Pages 31-36
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The Gross-Pitaevskii Regime
Pages 37-56
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Table of contents (7 chapters)
- Download Preface 1 PDF (38.6 KB)
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- Download Table of contents PDF (35.6 KB)
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Bibliographic Information
- Bibliographic Information
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- Book Title
- Effective Evolution Equations from Quantum Dynamics
- Authors
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- Niels Benedikter
- Marcello Porta
- Benjamin Schlein
- Series Title
- SpringerBriefs in Mathematical Physics
- Series Volume
- 7
- Copyright
- 2016
- Publisher
- Springer International Publishing
- Copyright Holder
- The Author(s)
- eBook ISBN
- 978-3-319-24898-1
- DOI
- 10.1007/978-3-319-24898-1
- Softcover ISBN
- 978-3-319-24896-7
- Series ISSN
- 2197-1757
- Edition Number
- 1
- Number of Pages
- VII, 91
- Topics
