Combined mean-field and semiclassical limits of large fermionic systems

TL;DR

This paper investigates the semiclassical limit of large fermionic systems, demonstrating that the Husimi measure converges to the Vlasov equation in a three-dimensional setting.

Contribution

It introduces a novel approach using Husimi measures and hierarchy estimates to connect quantum fermionic dynamics with classical Vlasov equations.

Findings

Husimi measure converges to Vlasov solution in the semiclassical limit

Established uniform estimates for hierarchy remainder terms

Proved weak compactness of Husimi measures in large fermionic systems

Abstract

We study the time dependent Schr\"odinger equation for large spinless fermions with the semiclassical scale $\hbar = N^{-1/3}$ in three dimensions. By using the Husimi measure defined by coherent states, we rewrite the Schr\"odinger equation into a BBGKY type of hierarchy for the k particle Husimi measure. Further estimates are derived to obtain the weak compactness of the Husimi measure, and in addition uniform estimates for the remainder terms in the hierarchy are derived in order to show that in the semiclassical regime the weak limit of the Husimi measure is exactly the solution of the Vlasov equation.