Phase Transitions Without Thermodynamic Limit,The Crucial R\^ole of   Possible and Impossible Fluctuations, The Treatment of Inhomogeneous Scenaria   in the Microcanonical Ensemble
  Microcanonical thermodynamics (MCTh) is contrasted to canonical thermodynamics (CTh). At phase transitions of 1.order the two ensembles are NOT equivalent even in the thermodynamic limit . Energy fluctuations do not vanish and phase separations are suppressed in CTh. A proper treatment of fluctuations is neccessary. MCTh allows to address even isolated small systems where phase transitions can be clearly classified into first order and continuous ones. The microcanonical caloric curve T(E) determines the transition temperature, latent heat AND surface entropy/tension. For systems of ca.1000 Na-, K-, or Fe-atoms at 1 atm. all 3 quantities can be calculated. The three parameters approach with rising size the known bulk values. There is nothing that demands the use of the thermodynamic limit. Within microcanonical thermodynamics of finite systems there are fundamental differences between conserved extensive variables and ensemble related ones like entropy, temperature and pressure. This is discussed in detail. 
D. H. E. Gross (Hahn-Meitner Institut,Berlin)