Phase Transitions in "Small" Systems - A Challenge for Thermodynamics

Traditionally, phase transitions are defined in the thermodynamic limit only. We propose a new formulation of equilibrium thermo-dynamics that is based entirely on mechanics and reflects just the {\em geometry and topology} of the N-body phase-space as function of the conserved quantities, energy, particle number and others. This allows to define thermo-statistics {\em without the use of the thermodynamic limit}, to apply it to ``Small'' systems as well and to define phase transitions unambiguously also there. ``Small'' systems are systems where the linear dimension is of the characteristic range of the interaction between the particles. Also astrophysical systems are ``Small'' in this sense. Boltzmann defines the entropy as the logarithm of the area $W(E,N)=e^{S(E,N)}$ of the surface in the mechanical N-body phase space at total energy E. The topology of S(E,N) or more precisely, of the curvature determinant $D(E,N)=\partial^2S/\partial E^2*\partial^2S/\partial N^2-(\partial^2S/\partial E\partial N)^2$ allows the classification of phase transitions {\em without taking the thermodynamic limit}. The topology gives further a simple and transparent definition of the {\em order parameter.} Attention: Boltzmann's entropy S(E) as defined here is different from the information entropy and can even be non-extensive and convex.Comment: 8 pages, 4 figures, Invited paper for CRIS200

What can nuclear collisions teach us about the boiling of water or the formation of multi-star systems ?

Phase transitions in nuclei, small atomic clusters and self-gravitating systems demand the extension of thermo-statistics to ``Small'' systems. The main obstacle is the thermodynamic limit. It is shown how the original definition of the entropy by Boltzmann as the volume of the energy-manifold of the N-body phase space allows a {\em geometrical} definition of the entropy as function of the conserved quantities. Without invoking the thermodynamic limit the whole ``zoo'' of phase transitions and critical points/lines can be unambiguously defined. The relation to the Yang--Lee singularities of the grand-canonical partition sum is pointed out. It is shown that just phase transitions in non-extensive systems give the complete set of characteristic parameters of the transition {\em including the surface tension.} Nuclear heavy-ion collisions are an experimental playground to explore this extension of thermo-statisticsComment: Invited talk for Bologna 2000, 8 pages, 3 figures.ep

Remarkable long-range-systematic in the binding energies of alpha-nuclei II

In this letter I present further data that show the remarkable evidence for the existence of an alpha-cluster stucture in the ground states of even-even N=Z nuclei. Such a remarkable systematic was observed 20 years ago for these nuclei at A<= 72 and is extended here up to A=100.Comment: 3 pages, 2 figure