  The most complicated phenomena of equilibrium statistics, phase separations
and transitions of various order and critical phenomena, can clearly and
sharply be seen even for small systems in the topology of the curvature of the
microcanonical entropy $S_{B}(E,N)=\ln[W(E,N)]$ as function of the conserved
energy, particle number etc.. Also the equilibrium of the largest possible
interacting many-body systems like self-gravitating systems is described by the
topology of the entropy surface $S_{B}(E,N,L)$ where $L$ is the angular
momentum. Conventional (canonical) statistical mechanics describes phase
transitions only in the ``thermodynamic limit''(homogeneous phases of
``infinite'' systems interacting with short-range interactions). In this paper
I present two examples of phase transitions of first order: the liquid to gas
transition in a small atomic cluster and the condensation of a rotating
self-gravitating system into single stars or into multi-star systems like
double stars and rings. Such systems cannot be addressed by ordinary canonical
thermo-statistics. I also give a geometric illustration how an initially
non-equilibrized ensemble approaches the microcanonical equilibrium
distribution.
