Derivation of Vlasov equations for multi-species fermions

We explore a system of two-species fermions with N particles. Through this exploration, we establish a two-species Husimi measure and construct a two-species Bogoliubov-Born-Green-Kirkwood-Yvon hierarchy hierarchy with error terms. We provide proofs of the smallness of these errors in the semiclassical and mean-field limits, fixing ℏ = N^−1/3. After that, we rigorously establish the uniqueness of this hierarchy. This allows us to conclude that the two-species Husimi measure from the solution of N-particle Schrödinger equation approximates the solution of the Vlasov equation for two-species systems.