Phase Transitions in Finite Systems using Information Theory

Abstract

In this paper, we present the issues we consider as essential as far as the statistical mechanics of finite systems is concerned. In particular, we emphasize our present understanding of phase transitions in the framework of information theory. Information theory provides a thermodynamically‐consistent treatment of finite, open, transient and expanding systems which are difficult problems in approaches using standard statistical ensembles. As an example, we analyze the problem of boundary conditions, which in the framework of information theory must also be treated statistically. We recall that out of the thermodynamical limit the different ensembles are not equivalent and in particular they may lead to dramatically different equations of state, in the region of a first order phase transition. We recall the recent progresses achieved in the understanding of first‐order phase transition in finite systems: the equivalence between the Yang‐Lee theorem and the occurrence of bimodalities in the intensive ensemble and the presence of inverted curvatures of the thermodynamic potential of the associated extensive ensemble. We come back to the concept of order parameters and to the role of constraints on order parameters in order to predict the expected signature of first‐order phase transition: in absence of any constraint (intensive ensemble) bimodality of the event distribution is expected while an inverted curvature of the thermodynamic potential is expected at a fixed value of the order parameter (extensive ensemble) in between the phases (co‐existence zone). 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.