We study the dynamics of many-body Fermi systems, for a class of initial data
which are close to quasi-free states exhibiting a nonvanishing pairing matrix.
We focus on the mean-field scaling, which for fermionic systems is naturally
coupled with a semiclassical scaling. Under the assumption that the initial
datum enjoys a suitable semiclassical structure, we give a rigorous derivation
of the time-dependent Hartree-Fock-Bogoliubov equation, a nonlinear effective
evolution equation for the one-particle density matrix of the system, as the
number of particles goes to infinity. Our result holds for all macroscopic
times, and provides bounds for the rate of convergence.Comment: 43 page