Micro-canonical statistical mechanics of some non-extensive systems

Non-extensive systems are not allowed to go to the thermodynamic limit. Therefore statistical mechanics has to be reformulated without invoking the thermodynamical limit. That is, we have to go back to pre-Gibbsian times. It is shown that Boltzmannʹs mechanical definition of entropy S as a function of the conserved “extensive” variables energy E, particle number N, etc., allows the description of even the most sophisticated cases of phase transitions unambiguously for “small” systems like nuclei, atomic clusters, and self-gravitating astrophysical systems. The rich topology of the curvature of S(E,N) shows the whole “zoo” of transitions: transitions of first 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.