The challenges of finite-system statistical mechanics

In this paper, we review the main challenges associated with the statistical mechanics of finite systems, with a particular emphasis on the present understanding of phase transitions in the framework of information theory. We show that this is a very powerful formalism allowing to treat in a thermodynamically consistent way many difficult problems in the statistical treatment of finite, open, transient and expanding systems. The first point we analyze is the problem of boundary conditions, which in the framework of information theory must also be treated statistically. We recall that the different ensembles do not lead to the same equation of states, in particular in the region of a first-order phase transition, and we stress the fact that different statistical ensembles may be relevant to heavy-ion physics depending upon the actual experimental conditions. Finally, we present a coherent description of first-order phase transitions demonstrating the equivalence between the Yang-Lee theorem, the occurrence of bimodalities in the intensive ensemble and the presence of inverted curvatures of the thermodynamic potential of the extensive ensemble. We stress that this discussion is not restricted to the possible occurrence of negative specific heat, but can also include negative compressibilities and negative susceptibilities, and in fact any curvature anomaly of the thermodynamic potential. Since the relevant entropy surface explored in nuclear multifragmentation is not yet well understood and largely debated in the community, the experimental evidence of new thermodynamic anomalies is one of the important challenges of future heavy-ion experiments