Semiclassical limit to the Vlasov equation with inverse power law potentials

Abstract

We consider mixed quasi-free states describing $N$ fermions in the mean-field limit. In this regime, the time evolution is governed by the nonlinear Hartree equation. In the large $N$ limit, we study the convergence towards the classical Vlasov equation. Under integrability and regularity assumptions on the initial state, we prove strong convergence in trace and Hilbert-Schmidt norm and provide explicit bounds on the convergence rate for a class of singular potentials of the form $V(x)=|x{|}^{-\alpha }$, for $\alpha \in (0,1/2)$.

arXiv comment45 pages

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• Enhanced Stability in Quantum Optimal Transport Pseudometrics: From Hartree to Vlasov-Poisson
• Derivation of Vlasov equations for multi-species fermions
• Effective Dynamics of Interacting Fermions from Semiclassical Theory to the Random Phase Approximation