Benedikter N., Porta M., Schlein B. Effective Evolution Equations from Quantum Dynamics

Springer International Publishing, 2016 — 91p. — (SpringerBriefs in Mathematical Physics 7) — ISBN: 978-3-319-24898-1 (eBook), 978-3-319-24896-7 (Softcover).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.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