A fundamental issue lying at the foundation of classical statistical mechanics is the determination of the collision boundary conditions that characterize the dynamical evolution of multi-particle probability density functions (PDF) and are applicable tosystems of hard-spheres undergoing multiple elastic collisions. In this paper it is proved that, when the deterministic N-body PDF is included in the class of admissible solutions of the Liouville equation, the customary form of collision boundary conditions adopted in previous literature becomes physically inconsistent and must actually be replaced by suitably modified collision boundary conditions.