Reconciliation of Statistical Mechanics and Astro-Physical Statistics.   The errors of conventional canonical thermostatistics
  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. 
D.H.E.Gross