  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 $\sim 100$
atoms by the shape of the microcanonical caloric equation of state $<T(E/N)>$
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. 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 $\Delta s_{surf}$ can very well be determined in relatively small
systems like clusters by MT. 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.
