A two-dimensional class of mean-field models that may serve as a minimal model to study the properties of long-range systems in two space dimensions is considered. The statistical equilibrium mechanics is derived in the microcanonical ensemble using Monte Carlo simulations for different...

We study the mean-field φ4 model in an external magnetic field in the microcanonical ensemble using two different methods. The first one is based on Rugh's microcanonical formalism and leads to express macroscopic observables, such as temperature, specific heat, magnetization and...

Ensemble inequivalence has been observed in several systems. In particular, it has been recently shown that negative specific heat can arise in the microcanonical ensemble in the thermodynamic limit for systems with long-range interactions. We display a connection between such behaviour and a...

We present a complete analytical solution of a system of Potts spins on a random k-regular graph in both the canonical and microcanonical ensembles, using the Large Deviation Cavity Method (LDCM). The solution is shown to be composed of three different branches, resulting in a non-concave...

We consider systems in which the canonical partition function can be expressed as the integral in an n-dimensional space (the order parameter space) of a function that also depends parametrically on the number N of degrees of freedom and on the inverse temperature β. We show how to compute,...

We review simple aspects of the thermodynamic and dynamical properties of systems with long-range pairwise interactions (LRI), which decay as 1/rd+σ at large distances r in d dimensions. Two broad classes of such systems are discussed. (i) Systems with a slow decay of the interactions, termed...

The relation between thermodynamic phase transitions in classical systems and topology changes in their state space is discussed for systems in which equivalence of statistical ensembles does not hold. As an example, the spherical model with mean field-type interactions is considered. Exact...

The effect of nearest-neighbor coupling on the thermodynamic and dynamical properties of the ferromagnetic Hamiltonian mean field model (HMF) is studied. For a range of antiferromagnetic nearest-neighbor coupling, a canonical first-order transition is observed, and the canonical and...