A study of Entropies and Non-linear Constraints in Long Range Interacting Systems

The statistical basis for entropy has been laid by Boltzmann and Planck giving
$S_{N}=k_{B}\ln{\Omega}$, where $S_{N}$ is the total thermodynamic entropy of the
system, $N$ the number of entities, $\Omega$ the statistical weight or number of
possible realisations (e.g., microstates) of the system, of equal
probabilities, and $k_{B}$ is the Boltzmann constant. For a discrete system
one may also write $S=-k_{B} \sum p_{i}\ln{p_{i}}$. $p_{i}$ is the probability of occurrence of the $i$-$th$
distinguishable outcome or state, from a total of $s$ such states.
In a previous paper those authors provide a natural extension of the
Boltzmann counting method in order to obtain generalized entropies, which
leads to a statistical interpretation based on the occupational statistics of
a stochastic process.
In this paper we impose non-linear constraints to the entropy function. In particular we explore these in systems with long range interaction.
The system studied is the Hamiltonian Mean Field (HMF) model. To study the behavior of this system, simulations using Microcanonical Monte Carlo (MMC) method was used. At first, we observed the behavior without non-linear constraints, only the conservation of energy and particles. Then, we study the behavior of the system with addition of the non-linear constraint.
With addition of the non-linear constraint the system remained in the phase given by the initial conditions, i.e. if the system began magnetized it remained magnetized at high energies and vice versa.