Classical equilibrium thermostatistics, “Sancta sanctorum of Statistical Mechanics” from nuclei to stars

Equilibrium statistics of Hamiltonian systems is correctly described by the microcanonical ensemble. Classically this is the manifold of all points in the N-body phase space with a given total energy. Due to Boltzmann–Planck's principle, eS=tr(δ(E−H)), its geometrical size is related to the entropy S(E,N,V,…). This definition does not invoke any information theory, no thermodynamic limit, no extensivity, and no homogeneity assumption. Therefore, it describes the equilibrium statistics of extensive as well of non-extensive systems. Due to this fact it is the fundamental definition of any classical equilibrium statistics. It addresses nuclei and astrophysical objects as well. S(E,N,V,…) is multiply differentiable everywhere, even at phase transitions. All kind of phase transitions can be distinguished sharply and uniquely for even small systems. In contrast to the canonical theory, what is even more important, is that the region of phase space which corresponds to phase separation is accessible, where the most interesting phenomena occur. No deformed q-entropy is needed for equilibrium. Boltzmann–Planck is the only appropriate statistics independent of whether the system is small or large, whether the system is ruled by short- or long range forces.