On the Microscopic Foundation of Thermo-Statistics

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.