Semiclassical Limit of the Bogoliubov–de Gennes Equation

In this paper, we rewrite the time-dependent Bogoliubovx2013de Gennes equation in an appropriate semiclassical form and establish its semiclassical limit to a two-particle kinetic transport equation with an effective mean-field background potential satisfying the one-particle Vlasov equation. Moreover, for some semiclassical regimes, we obtain a higher-order correction to the two-particle kinetic transport equation, capturing a nontrivial two-body interaction effect. The convergence is proven for C² interaction potentials in terms of a semiclassical optimal transport pseudo-metric. Furthermore, combining our current results with the results of Marcantoni et al. arXiv: 2310. 15280, we establish a joint semiclassical and mean-field approximation of the dynamics of a system of spin-12 Fermions by the Vlasov equation in some negative order Sobolev topology.