First-principle proof of the modified collision boundary conditions for the hard-sphere system

• Statistical treatment of the collisional dynamics of the hard-sphere system.
• Inclusion of distributions as admissible solutions of the Liouville equation.
• Collision boundary conditions used in past literature are physically inconsistent.
• New modified collision boundary conditions are proposed and proved to apply.
• The result is relevant for the study of dense or locally-dense gases and fluids.

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 to systems 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.