  Conventional thermo-statistics address infinite homogeneous systems within
the canonical ensemble. (Only in this case this is equivalent to the
fundamental microcanonical 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. 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. As no
extensivity, no thermodynamic limit, no concavity, no homogeneity is needed, it
also applies to astro-physical systems. 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 the present paper the necessary appearance of a convex entropy S(E) and
negative heat capacity at phase separation in small as well macroscopic systems
independently of the range of the force is pointed out. Thus the old puzzle of
stellar statistics is finally solved, the appearance of negative heat capacity
which is forbidden and cannot appear in the canonical formalism.
