Effective Equations for Fermionic Systems

Descripción del proyecto

Investigar la dinámica no lineal de grandes sistemas fermiónicos

Los sistemas fermiónicos desempeñan un papel importante en la descripción de las moléculas y la materia condensada. Su evolución temporal viene determinada por la ecuación de Schrödinger que, sin embargo, es muy difícil de analizar para sistemas grandes con muchas partículas. El equipo del proyecto EFFECT, financiado por las Acciones Marie Skłodowska-Curie, tiene como objetivo mejorar la comprensión de la dinámica de no equilibrio de los grandes sistemas fermiónicos y su interacción con los campos electromagnéticos cuantizados. El equipo del proyecto tiene previsto desarrollar nuevas herramientas matemáticas para abordar estos grandes sistemas mediante ecuaciones de evolución efectivas más sencillas para sistemas fermiónicos a temperaturas cero y finitas.

Objetivo

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.