  Conventional thermo-statistics address infinite homogeneous systems within
the canonical ensemble. However, some 170 years ago the original motivation of
thermodynamics was the description of steam engines, i.e. boiling water. Its
essential physics is the separation of the gas phase from the liquid. Of
course, boiling water is inhomogeneous and as such cannot be treated by
conventional thermo-statistics. Then it is not astonishing, that a phase
transition of first order is signaled canonically by a Yang-Lee singularity.
Thus it is only treated correctly by microcanonical Boltzmann-Planck
statistics. This is elaborated in the present paper. It turns out that the
Boltzmann-Planck statistics is much richer and gives fundamental insight into
statistical mechanics and especially into entropy. This can be done to a far
extend rigorously and analytically. The deep and essential difference between
``extensive'' and ``intensive'' control parameters, i.e. microcanonical and
canonical statistics, is exemplified by rotating, self-gravitating systems. In
this paper the necessary appearance of a convex entropy $S(E)$ and the negative
heat capacity at phase separation in small as well macroscopic systems
independently of the range of the force is pointed out. The appearance of a
critical end-point for the liquid-gas transition in the $p-E$ or $V-E$ phase
diagram can be easily explained as well the non-existence of a critical
end-point of the solid-liquid transition.
