6 - Gibbsian formalism and metastates

Summary

Longtemps les objets dont s'occupent les mathématiciens étaient pour la pluspart mal définis; on croyait les connaître, parce qu'on se les représentait avec le sens ou l'imagination; mais on n'en avait qu'une image grossière et non une idée précise sur laquelle le raisonnement pût avoir prise.

Henri Poincaré, La valeur de la science.

We now turn to the main topic of this book, disordered systems. We split this into two parts, treating in turn lattice models and mean-field models. From the physical point of view, the former should be more realistic and hence more relevant, so it is natural that we present the general mathematical framework in this context. However, the number of concrete problems one can to this day solve rigorously is quite limited, so that the examples we will treat can mostly be considered as random perturbations of the Ising model. In the second part we will be able to look, in a simplified setting, at more complex, genuinely random models, that rather surprisingly will turn out to produce fascinating mathematics, but also lead to applications that are beyond the standard scope of physics.

Introduction

We have seen that statistical mechanics is a theory that treats the dynamical degrees of freedom of a large system as random variables, distributed according to the Gibbs measure.