Effective Equations for Light-Matter Interactions and Fermionic Systems at Finite Temperature

• Applicant: Professor Dr. Nikolai Leopold
• Subject Area: Mathematics
• Term: since 2026
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 580841385

This project aims to gain a deeper understanding of the non-equilibrium dynamics of fermionic systems. Fermions are the fundamental building blocks of matter, and understanding their behavior is crucial for describing atoms and molecules. The time evolution of fermionic systems is governed by the quantum mechanical Schrödinger equation, which, however, cannot be solved analytically or numerically for many-particle systems. Therefore, effective equations are used, which approximate the full quantum dynamics but are easier to analyze. The main part of the project is concerned with the mathematically rigorous validation of such equations for systems of many fermions at finite temperature. Specifically, the Maxwell-Schrödinger equations will be derived from the Pauli-Fierz dynamics -- a mathematically rigorous model of non-relativistic quantum electrodynamics -- in the semiclassical mean-field limit for fermions in the grand canonical ensemble. Combined with previous results by the applicant and a co-author, this enables, for the first time, a rigorous derivation of the regularized Vlasov-Maxwell equations with a reasonable rate of convergence. Moreover, a new method will be developed within the first-quantization formalism to rigorously derive mean-field and kinetic equations from the quantum dynamics of many fermions in the canonical and microcanonical ensembles. As an initial subproject, the Newton-Maxwell equations will be derived from the Pauli–Fierz dynamics in the classical limit with an explicit rate of convergence. The project will deepen the understanding of how the Maxwell equations emerge as an effective theory of quantum electrodynamics and will provide new tools for the effective description of fermionic systems at finite temperature. The methods developed are expected to be valuable also for studies of systems with singular interactions and bosonic systems at finite temperatures.

• DFG Programme: Research Grants