**1. Long-range interactions**

The description of systems with long-range interactions is relevant to statistical physics because we find appealing properties that deserve to be studied in detail. In this line, some variations of the Ising model that involves not only first neighbors, but also distant neighbors are employed; for instance, the Hamiltonian Mean Field (HMF), and the recently introduced dipole-type Hamiltonian Mean Field (d-HMF) model. We emphasize that the Ising model is a recurrent tool to study magnetic properties and the statistical behaviors of many-body systems in the broadest context [1–5].

In concern about long-range interactions, there are various challenges to face; these are related to the dynamics, size of the systems, and the theoretical framework to explain the behavior of systems, among others. In this type of system, we have typical consequences such as the loss of additivity and extensivity. The loss of additivity takes place for ensembles of interacting particles that cannot trivially separate into independent subsystems, which is explained by the presence of underlying interactions or correlation effects, whose characteristic lengths are comparable or more significant than the system linear size [6–8]. Additionally, the loss of extensivity frequently accompanies the loss of additivity [5, 9].

If we have a system composed of N spins that interact one to each other as a power law of inter-particle distance, the total energy of the system is E. The system divided into two subsystems, 1 and 2, both composed of N/2 particles with energies *E*1 and *E*<sup>2</sup> , respectively. If the spins in subsystem 1 are up, and those in the subsystem 2 are down, the energies satisfy *E*<sup>1</sup> = *E*<sup>2</sup> . Since the sum of energies, *E E* 1 2 + is not equal to the total energy E; the system is nonadditive. In general, *EE E* < +1 2 . Besides, the extensivity ( *E N*/ →constant) is another fundamental property to consider in this kind of analysis. The recent literature shows a way to recover this property through Kac's prescription that has a standard thermodynamic structure because it preserves the Euler and Gibbs-Duhem relations. Therefore, this procedure allows us to recover the extensivity, while the loss of additivity remains because of the long-range interactions [5, 9].

At equilibrium, an analytical procedure leads to solving the problem for obtaining the magnetization, the inverse temperature, the specific heat, in the canonical ensemble. Also, it is possible to get the microcanonical entropy. The caloric curve, ascertained using both, exactly coincides. At this stage, we emphasize that the solution to this system with long-range interactions becomes analytical, which solutions are not abundant in statistical physics. Nevertheless, out of equilibrium, we have data from simulations obtained from carrying out molecular dynamics.

The evolution of simulations permits us to observe several properties that are not possible to identify with any theoretical description. The interpretation comes from perceiving regularities in the dynamics of systems as HMF and d-HMF models.
