  In this paper, we study a cold gas of $N \gg 1$ weakly interacting fermions.
We describe the time evolution of the momentum distribution of states close to
the Fermi ball by simultaneously analyzing the dynamical behavior of excited
particles and holes. Our main result states that, for small values of the
coupling constant, and for appropriate initial data, the effective dynamics of
the above system is driven by an energy-mollified quantum Boltzmann collision
operator, plus a an interaction term with virtual bosonized particle-hole pairs
around the Fermi surface.
