Effective Equations for Fermionic Systems

Projektbeschreibung

Auslotung der nicht linearen Dynamik umfassender fermionischer Systeme

Fermionische Systeme spielen eine wichtige Rolle bei der Beschreibung von Molekülen und kondensierter Materie. Ihre Entwicklung im Laufe der Zeit wird durch die Schrödingergleichung beschrieben, deren Analyse ist allerdings für große Systeme mit vielen Teilchen sehr schwierig. Das im Rahmen der Marie-Skłodowska-Curie-Maßnahmen finanzierte Projekt EFFECT hat sich vorgenommen, das Verständnis der Nichtgleichgewichts-Dynamik umfassender fermionischer Systeme und ihrer Interaktion mit quantisierten elektromagnetischen Feldern zu vertiefen. Das Projekt zielt auf die Entwicklung neuer mathematischer Hilfsmittel ab, um derart große Systeme durch einfachere effektive Entwicklungsgleichungen für fermionische Systeme bei Nulltemperatur und endlichen Temperaturen zu approximieren.

Ziel

The goal of this project is to substantially improve the understanding of the non-equilibrium dynamics of large fermionic systems and their interaction with the quantized electromagnetic field. Fermionic systems play a significant role in the description of molecules and condensed matter. Their time evolution is determined by the Schrödinger equation which, however, is very challenging to analyze for large systems with many particles. For this reason simpler effective equations are used to approximately predict the time evolution. These are easier to investigate but less exact. In physics, effective equations are derived by heuristic arguments. Beyond that, a mathematical analysis is essential to prove the range of validity of the applied approximation. In the scope of this project new mathematical tools will be developed to rigorously derive effective evolution equations for fermionic systems at zero and finite temperature. The Hartree-Fock equation with Coulomb potential will be derived from the Schrödinger equation in a many-fermion mean-field limit which is coupled to a semiclassical limit. In the same scaling limit the use of the (fermionic) Maxwell-Schrödinger equations as approximate time evolution of the Pauli-Fierz Hamiltonian will be justified. Moreover, it will be proven that the quantum fluctuations around the effective equations are described by Bogoliubov theory. Explicit estimates for the error caused by the approximation will be provided. In total, this will enhance the understanding about the creation of correlations among fermions and the emergence of classical field theories from quantum field theories. The derivations are long outstanding and there is an extensive need for new mathematical methods in semiclassical analysis and many-body quantum mechanics. It is expected that the new techniques will also have a strong impact on studies about dilute Bose gases at positive temperature and fermionic systems in the kinetic regime.