Microscopic statistical basis of classical Thermodynamics of finite systems

Heat can flow from cold to hot at any phase separation. Therefore Lynden-Bell's gravo-thermal catastrophe must be reconsidered. The original objects of Thermodynamics, the separation of phases at first order phase transitions, like boiling water in steam engines, are not described by a single canonical ensemble. Inter-phase fluctuations are not covered. The basic principles of statistical mechanics, especially of phase transitions have to be reconsidered without the use of the thermodynamic limit. Then thermo-statistics applies also to nuclei and large astronomical systems. A lot of similarity exists between the accessible phase space of fragmenting nuclei and inhomogeneous multi stellar systems.