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The volume W of the accessible N-body phase space and its dependence on the total energy is directly calculated. The famous Boltzmann relation S = k * ln(W ) defines microcanonical thermodynamics (MT). We study how phase transitions appear in MT. Here we first develop the thermodynamics of microcanonical phase transitions of first and second order in systems which are thermodynamically stable in the sense of van Hove. We show how both kinds of phase transitions can unambiguously be identified in relatively small isolated systems of ∼ 100 atoms by the shape of the microcanonical caloric equation of state < T (E/N ) > and not so well by the coexistence of two spatially clearly separated phases. I.e.within microcanonical thermodynamics one does not need to go to the thermodynamic limit in order to identify phase transitions. In contrast to ordinary (canonical) thermodynamics of the bulk microcanonical thermodynamics (MT) gives an insight into the coexistence region. Here the form of the specific heat c(E/N ) connects transitions of first and second order in a natural way. The essential three parameters which identify the transition to be of first order, the transition temperature T tr , the latent heat q lat , and the interphase surface entropy ∆s surf can very well be 1 determined in relatively small systems like clusters by MT. It turns out to be essential whether the cluster is studied canonically at constant temperature or microcanonically at constant energy. Especially the study of phase separations like solid and liquid or, as studied here, liquid and gas is very natural in the microcanonical ensemble, whereas phase separations become exponentially suppressed within the canonical description. The phase transition towards fragmentation is introduced. The general features of MT as applied to the fragmentation of atomic clusters are discussed. The similarities and differences to the boiling of macrosystems are pointed out.

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We discuss the role and the treatment of polarization effects in many-body systems of charged conducting clusters and apply this to the statistical fragmentation of Naclusters. We see a first order microcanonical phase transition in the fragmentation of Na Z+ 70 for Z = 0 to 8. We can distinguish two fragmentation phases, namely evaporation of large particles from a large residue and a complete decay into small fragments only. Charging the cluster shifts the transition to lower excitation energies and forces the transition to disappear for charges higher than Z = 8. At very high charges the fragmentation phase transition no longer occurs because the cluster Coulomb-explodes into small fragments even at excitation energy * = 0. PACS: 65.50.+m; 64.70.Fx