Macroscopic irreversibility and decay to kinetic equilibrium for classical hard-sphere systems

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.