Time-dependent effective one-particle equations for large interacting fermionic systems

In the quantum scale, dynamics of many-particles in a system are governed by Schrödinger's quantum systems. However, it is computationally costly to run a simulation if the system consists of a large quantity of particles, which is often the case. Therefore, the method of deriving an effective equation that captures the macroscopic behavior of the particles is of high interest in the mathematical physics community. In this doctoral thesis, we focus on the derivation of an effective time-dependent equation of a macroscopic system consisting of fermions, a type of particle that adhere to Pauli's exclusion principle. In particular, we derive a Vlasov hierarchy from the Schrödinger system with respect to the Husimi measure. Then, we show that under regularized interaction potential, the quantum system converges to Vlasov equation in terms semiclassical limit by using the BBGKY method and Bogoliubov transformation independently. Furthermore, we extend the convergence result to the Vlasov-Poisson equation by considering a truncated Coulomb potential