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This chapter begins with a rapid introduction to the definition and construction of the statistical ensembles, following the lines that are usually offered in the basics courses on statistical mechanics. The reader is assumed to be already acquainted with thermodynamics and basic statistical mechanics, but the introduction is self-contained. It is then shown that with short-range interactions the ensembles are physically equivalent, even in the presence of phase transitions. This is done by introducing the notions of concave functions, of stable and tempered potentials, and the Legendre-Fenchel transform of thermodynamic functions. The concavity of the entropy is at the basis of ensemble equivalence. Finally, the concepts of microstate and macrostate are analysed; they are very important for the study of ensemble equivalence or inequivalence in long-range systems.

This chapter deals with the peculiarities that are present in the statistical mechanics treatment and in the equilibrium properties of system with long-range interactions. In dimension d they are characterized at large distance r by a decay slower than 1/rd. The important concepts of additivity and extensivity are analysed. It is shown that long-range systems are intrinsically nonadditive: the interaction energy between macroscopic subsystems is not negligible with respect to the total energy of the system. This can often give rise to ensemble inequivalence. In particular, also with the help of a simple model, the possibility to have convex regions in the entropy is demonstrated. These regions can cause a negative specific heat in the microcanonical ensemble, while in the canonical ensemble this quantity is always positive. A negative magnetic susceptibility is possible, as well, in the microcanonical ensemble.

This chapter is devoted to the study of a very useful mathematical tool for the computation of the entropy in long-range systems. The method is based on large deviation theory, and it is very well suited for purely mean-field systems, in which the interaction does not decay at all with the distance. However, it can be applied also to the other long-range systems. The three steps of the method are described: the expression of the energy in terms of global variables, the mean fields; the computation of the entropy of the macrostates, defined by given values of the mean fields; the computation of the entropy, maximizing on the macrostates. Finally, the method is illustrated by applying it to two simple models.

This chapter aims at showing that the features occurring in mean-field models, described in the previous chapters, can be found also in the other long-range systems. The first four sections are dedicated to generalizations of the models of chapter 4, in which either the mean-field interaction is augmented with a nearest neighbour interaction, or it is replaced by a slowly decaying interaction. It is shown that the long-range characteristics of the associated mean-field models are preserved, and in addition ensemble inequivalence, microcanonical negative specific heat and ergodicity breaking are induced in some cases. The final section introduces the dipolar interaction, a marginal long-range system. Dipolar systems are treated in details in chapter 15, and in this chapter few relevant properties are presented, focussing in particular on elongated ferromagnets and on ergodicity breaking.

We study the distribution of the resistance fluctuations of biased resistor networks in nonequilibrium steady states. The stationary conditions arise from the competition between two stochastic and biased processes of breaking and recovery of the elementary resistors. The fluctuations of the network resistance are calculated by Monte Carlo simulations which are performed for different values of the applied current, for networks of different size and shape and by considering different levels of intrinsic disorder. The distribution of the resistance fluctuations generally exhibits relevant deviations from Gaussianity, in particular when the current approaches the threshold of electrical breakdown. For two-dimensional systems we have shown that this non-Gaussianity is in general related to finite size effects, thus it vanishes in the thermodynamic limit, with the remarkable exception of highly disordered networks. For these systems, close to the critical point of the conductor-insulator transition, non-Gaussianity persists in the large size limit and it is well described by the universal Bramwell-Holdsworth-Pinton distribution. In particular, here we analyze the role of the shape of the network on the distribution of the resistance fluctuations. Precisely, we consider quasi-one-dimensional networks elongated along the direction of the applied current or trasversal to it. A significant anisotropy is found for the properties of the distribution. These results apply to conducting thin films or wires with granular structure stressed by high current densities., 8 pages, 4 figures. Invited talk at the 18-th International Conference on Noise and Fluctuations, 19-23 September 2005, Salamanca