**2. Equilibrium**

As generally noticed, at low energy, a phase of a single cluster of particles appears; at high energy, a homogeneous phase emerges.

For theoretical and numerical modeling, we consider a system of N identical, coupled, dipole-type particles with a mass equal to 1. The dynamics evolves in a periodic cell described by a one-dimensional, dipole-type HMF model (d-HMF) [9, 10] given by

$$H = \frac{\mathbf{1}}{\mathbf{2}} \sum\_{i=1}^{N} p\_i^2 + \frac{\hat{\mathcal{A}}}{\mathbf{2}N} \sum\_{i \neq j}^{N} \left[ \cos \left( \theta\_i - \theta\_j \right) - \mathbf{g} \cos \theta\_i \cos \theta\_j - \Lambda\_{i,j} \right], \tag{1}$$

where the variable *pi* is the momentum of the particle *i*, and θ*i* is its corresponding angle of orientation (integer *i* ∈ [1,*N*] for the system size *N*). The parameter λstands for the coupling and ∆*i j* , suitably denotes the zero of the potential.

The equations of motion are derived by the following set:

$$p\_i = \frac{-\mathcal{E}}{2} \left( \mathbf{z}M\_x \sin \theta\_i - M\_y \cos \theta\_i \right),\tag{2}$$

where the components of the magnetization vector are defined as

$$\left(\mathcal{M}\_x, \mathcal{M}\_y\right) = \frac{1}{N} \sum\_{i \neq j}^{N} (\cos \theta\_i, \sin \theta\_i). \tag{3}$$

After deriving thermodynamic properties in both, the canonical and microcanonical ensembles, the caloric curves come from the following coincident result:

$$\text{tr}\,\beta = \frac{\mathbf{1}}{\left(\mathcal{E} + \mathcal{X}\left(m^\* + \Lambda/2\right)\right)},\tag{4}$$

combined with the extremal solution

$$m = \frac{I\_\circ \left(\text{2}\beta\lambda m\right)}{I\_\circ \left(\text{2}\beta\lambda m\right)},\tag{5}$$

**5**

**Figure 1.**

*Introductory Chapter: Statistical and Theoretical Considerations on Magnetism in Many-Body…*

comprehensive overview in his book [13]. The spherical model is an additional solv-

The statistical mechanics thoroughly explains the equilibrium, which is the statistical state reached in the long-lasting evolution. Nevertheless, in intermediate intervals of the evolution time, the dynamics is abnormal, and the description is not complete. Therefore, there are at least two regimes in the evolution of systems with long-range interactions: the equilibrium and the quasi-stationary states (QSS). In the first case, we have the Boltzmann-Gibbs statistics to describe the state of the systems entirely. In the second, it is an open problem that possesses several

Generally speaking, when the modeling neglects all of the spatial structure of the system but considers long-range interactions, mean-field approximations should be good alternatives to consider. The d-HMF model is curious because it neglects the structure of systems and space interactions, and only keeps the part

To identify the QSS, we search for intervals where the thermodynamic values keep constant. Therefore, **Figure 1(a)** illustrates two plateaus in the behavior of magnetization per particle as a function of time, before equilibrium. A complementary perspective of the first QSS given in [10] considers the shape of the mean kinetic energy per particle, aiding us to obtain the power-law duration depicted in **Figure 1(b)** that characterizes the nature of QSS. States defined by the plateaus of mean kinetic energy per particle are lower than the canonical temperature, whose

Finally, this system constitutes an example of a non-symmetric HMF that shows a phase transition, the appearance of a spontaneous magnetic ordering. The symmetry is compared to the HMF model that remains invariant under a typical rotational transformation, which is not valid for the d-HMF model, but presents a second-order phase transition; therefore, a possible application found in the literature is the phase transition of non-symmetric spin glasses. Theoretical background for systems out of equilibrium is not unique and fundamental questions related

*The magnetization per particle is depicted as a function of the time in (a). In (b), the relaxation time τ, of the first QSS that goes to the second QSS, is depicted as a function of 1/N in log–log scale, whose power law of the duration of the first QSS in terms of the system size is obtained from the behavior of the kinetic energy.*

related to orientations, but shows axiomatic and pertinent properties.

*DOI: http://dx.doi.org/10.5772/intechopen.93204*

**3. Quasi-stationary states**

able problem that includes long-range interactions.

theoretical attempts susceptible to improve them.

values only coincide in equilibrium [10].

where *Ik* (…) is the *k*th-order modified Bessel function, β is the inverse of temperature, ε is the energy per particle; *m* is the magnetization per particle. We analytically obtain the parameters that define the critical point, such as the critical inverse temperature *β*c = 1, critical internal energy *ε*c = 3/2, and critical magnetization *mc* = 0. The specific heat diverges with the temperature 1/ β → <sup>−</sup> 1 and remains in a constant value for 1/ β > 1 , which corresponds to an ideal gas in one dimension. Therefore, systems with magnetic behavior are recently studied in [11, 12], giving an approach to the understanding of magnetic materials.

As said before, this system denotes an analytically solvable many-body problem. This type of solutions is not common in physics and particularly in statistical mechanics, especially regarding long-range interactions [9]. Baxter gives a

comprehensive overview in his book [13]. The spherical model is an additional solvable problem that includes long-range interactions.
