  In this paper the problem is posed of the formulation of the so-called "ab
initio" approach to the statistical description of the Boltzmann-Sinai N-body
classical dynamical system (CDS) formed by identical smooth hard spheres. This
amounts to introducing a suitably-generalized version of the axioms of
Classical Statistical Mechanics. The latter involve a proper definition of the
functional setting for the N-body probability density function (PDF), so that
it includes also the case of the deterministic N-body PDF. In connection with
this issue, a further development concerns the introduction of modified
collision boundary conditions which differ from the usual ones adopted in
previous literature. Both features are proved to be consistent with the
validity of exact H-theorems for the N-body and 1-body PDFs respectively.
Consequences of the axiomatic approach which concern the conditions of validity
of the Boltzmann kinetic equation and the Boltzmann H-theorem are investigated.
In particular, the role of the modified boundary conditions is discussed. It is
shown that both theorems fail in the case in which the N-body PDF is identified
with the deterministic PDF. Finally, the issue of applicability of the Zermelo
and Loschmidt paradoxes to the "ab initio" approach presented here is
discussed.
