  Focusing on isolated macroscopic systems, described either in terms of a
quantum mechanical or a classical model, our two key questions are: In how far
does an initial ensemble (usually far from equilibrium and largely unknown in
detail) evolve towards a stationary long-time behavior ("equilibration")? In
how far is this steady state in agreement with the microcanonical ensemble as
predicted by Statistical Mechanics ("thermalization")? In the first part of the
paper, a recently developed quantum mechanical treatment of the problem is
briefly summarized, putting particular emphasis on the realistic modeling of
experimental measurements and non-equilibrium initial conditions. Within this
framework, equilibration can be proven under very weak assumptions about those
measurements and initial conditions, while thermalization still requires quite
strong additional hypotheses. In the second part, an analogous approach within
the framework of classical mechanics is developed and compared with the quantum
case. In particular, the assumptions to guarantee classical equilibration are
now rather strong, while thermalization then follows under relatively weak
additional conditions.
