Mean-field characterizations of first-order phase transitions:

Mean-field theory, dating back to Curie, Weiss and Ginzburg-Landau, is one of the most common approaches to the study of phase transitions in physical systems. The basic underlying idea is that the complicated interactions that each element of a complex system is subjected to by its neighbors can be replaced by the interaction with an effective (or mean) field. It has been a common belief that the "mean-field predictions" resulting from this approximation are fairly accurate in sufficiently high dimensions but, except perhaps for the results of Hara-Slade and Aizenman-Barsky-Fernandez concerning the critical exponents in high-dimensional percolation/Ising model, very little of this belief has been mathematically justified.

The problem has been addressed in the recent joint paper with L. Chayes. The basic thesis of this work is as follows: For any nearest neighbor ferromagnetic spin system, if the mean field theory indicates a first order phase transition, a corresponding transition will occur in this system provided the spatial dimension is sufficiently large. We also give bounds on the distance between the mean-field and actual transition. A work in progress shows that a similar statement will be true for sufficiently spread-out (but exponentially decaying) interactions once the spatial dimension is three or higher.

Relevant papers:

M. Biskup and L. Chayes, Rigorous analysis of discontinuous phase transitions via mean-field bounds, Commun. Math. Phys. 238 (2003), no. 1-2, 53-93.  pdf TeX

M. Biskup and L. Chayes, Mean-field driven first-order phase transitions in systems with long-range interactions, in preparation.