  In this paper the conditions are investigated for the occurrence of the
so-called macroscopic irreversibility property and the related phenomenon of
decay to kinetic equilibrium which may characterize the $1-$body probability
density function (PDF) associated with hard-sphere systems. The problem is set
in the framework of the axiomatic "ab initio" approach to classical statistical
mechanics recently developed [Tessarotto \textit{et al}., 2013-2017] and the
related establishment of an exact kinetic equation realized by Master equation
for the same kinetic PDF. As shown in the paper the task involves the
introduction of a suitable functional of the $1-$body PDF here identified with
the \textit{Master kinetic information}. The goal is to show that, provided the
same PDF is realized in terms of an arbitrary suitably-smooth particular
solution of the Master kinetic equation the two properties indicated above are
indeed realized and that the same functional is unrelated either with the
Boltzmann-Shannon entropy and the Fisher information.
