  Conventional thermo-statistics address infinite homogeneous systems within
the canonical ensemble. However, some 150 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
canonical 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 article. It turns out that the
Boltzmann-Planck statistics is much richer and gives fundamental insight into
statistical mechanics and especially into entropy. This can even be done to
some extend rigorously and analytically. The microcanonical entropy has a very
simple physical meaning: It measures the microscopic uncertainty that we have
about the system, i.e. the number of points in $6N$-dim phase, which are
consistent with our information about the system. It can rigorously be split
into an ideal-gas part and a configuration part which contains all the physics
and especially is responsible for all phase transitions. The deep and essential
difference between ``extensive'' and ``intensive'' control parameters, i.e.
microcanonical and canonical statistics, is exemplified by rotating,
self-gravitating systems.
