Typical Profiles of the Kac-Hopfield Model

Mean field models, random or not, are very important for explaining simply the general phenomenon of phase transitions. However, for random systems, in general, their analysis, as many of the contributions in this volume confirm, is not simple at all, a fact which may justify the amount of effort spent on them. In spite of all that, mean fields models, in many respects, are only poor caricatures of realistic systems1 and are unable to feature some of their most important aspects; in particular, in a phase-transition regime, they are unable to properly account for the phenomenon of phase separation, that is, the observed feature that states of the system with two or more phases coexist in separate regions of space. This deficiency manifests itself also in the fact that the canonical free energy is generally not a convex function of the order parameters, which in turn means that the usual formalism of thermodynamics cannot be immediately used (e.g., the isotherms are not monotone, thus cannot directly be used to determine the equations of state, and insisting on doing so would produce a totally unphysical effect, like regions of parameters where pressure is a decreasing function of density). This problem is solved by the Maxwell construction, by which free energy is simply replaced ad hoc by its convex hull.