Effective Evolution Equations from Quantum Dynamics

1. Front Matter

   Pages i-vii

   PDF

2. Introduction

   • Niels Benedikter, Marcello Porta, Benjamin Schlein

   Pages 1-6

3. Mean-Field Regime for Bosonic Systems

   • Niels Benedikter, Marcello Porta, Benjamin Schlein

   Pages 7-16

4. Coherent States Approach

   • Niels Benedikter, Marcello Porta, Benjamin Schlein

   Pages 17-29

5. Fluctuations Around Hartree Dynamics

   • Niels Benedikter, Marcello Porta, Benjamin Schlein

   Pages 31-36

6. The Gross-Pitaevskii Regime

   • Niels Benedikter, Marcello Porta, Benjamin Schlein

   Pages 37-56

7. Mean-Field Regime for Fermionic Systems

   • Niels Benedikter, Marcello Porta, Benjamin Schlein

   Pages 57-78

8. Dynamics of Fermionic Quasi-Free Mixed States

   • Niels Benedikter, Marcello Porta, Benjamin Schlein

   Pages 79-86

9. Back Matter

   Pages 87-91

   PDF

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.           

• 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
• Schrödinger Dynamics
• Slater Determinants

• Department of Mathematical Sciences, University of Copenhagen, Copenhagen, Denmark

  Niels Benedikter

• Institute of Mathematics, University of Zürich, Zürich, Switzerland

  Marcello Porta, Benjamin Schlein

Policies and ethics