Micro-canonical Statistical Mechanics of some Non-Extensive Systems

Abstract

Non-extensive systems do not allow to go to the thermodynamic limit. Therefore we have to reformulate statistical mechanics without invoking the thermodynamical limit. I.e. we have to go back to Pre-Gibbsian times. We show that Boltzmann's mechanical definition of entropy S as function of the conserved ``extensive'' variables energy E, particle number N etc. allows to describe even the most sophisticated cases of phase transitions unambiguously for ``small'' systems like nuclei, atomic clusters, and selfgravitating astrophysical systems: The rich topology of the curvature of S(E,N) shows the whole ``Zoo'' of transitions: transitions of 1.order including the surface tension at phase-separation, continuous transitions, critical and multi-critical points. The transitions are the ``catastrophes'' of the Laplace transform from the ``extensive'' to the ``intensive'' variables. Moreover, this classification of phase transitions is much more natural than the Yang-Lee criterion.