Modified BBGKY hierarchy for the hard-sphere system

In this paper a statistical approach is formulated for classical N-body systems formed by smooth hard spheres. Based on the emerging new axiomatic approach to Classical Statistical Mechanics recently developed, modified collision boundary conditions for the N-body probability density are introduced, which apply also to dense or locally dense hard-sphere systems. As a result, a modified form is determined for the BBGKY hierarchy, which is characterized by a new representation for the s-body collision operator. The same hierarchy, obtained here in differential form starting from the differential Liouville equation, is found to admit both stochastic and deterministic particular solutions. As an application, in the Boltzmann-Grad limit the hierarchy is shown to recover the ordinary Boltzmann equation holding in the case of rarefied gases. Comparison with literature and physical implications of the theory are pointed out.