  The effective approach to the foundation of the nonequilibrium statistical
mechanics on the basis of dynamics was formulated by Bogoliubov in his seminal
works. His ideas of reduced description were proved as very powerful and found
a broad applicability to quite general time-dependent problems of physics and
mechanics. In this paper we analyzed thoroughly the time evolution of open
systems in context of the nonequilibrium statistical operator method (NSO).
This method extends the statistical method of Gibbs to irreversible processes
and incorporates the ideas of reduced description. The purpose of the present
study was to elucidate the basic aspects of the NSO method and some few
selected approaches to the nonequilibrium statistical mechanics. The suitable
procedure of averaging (smoothing) and the notion of irreversibility were
discussed in this context. We were focused on the physical consistency of the
method as well as on its operational ability to emphasize and address a few
important reasons for such a workability.
