  The concept of time-coarsened density matrix for open systems has frequently
featured in equilibrium and non-equilibrium statistical mechanics, without
being probed as to the detailed consequences of the time averaging procedure.
In this work we introduce and prove the need for a selective and non-uniform
time-sampling, whose form depends on the properties (whether thermalized or
not) of the bath. It is also applicable when an open microscopic sub-system is
coupled to another {\it finite} system. By use of a time-periodic minimal
coupling model between these two systems, we present detailed quantitative
consequences of time coarsening, which include initial state independence of
equilibration, deviations from long term averages, their environment size
dependence and the approach to classicality, as measured by a Leggett-Garg type
inequality. An interacting multiple qubit model affords comparison between the
time integrating procedure and the more conventional environment tracing
method.
