Permutation entropy for the characterization of the attractive Hamiltonian mean-field model

The Hamiltonian mean-field (HMF) model is a long-range interaction system that exhibits quasistationary states (QSS), which persist for long times before reaching thermodynamic equilibrium. These states are traditionally characterized by homogeneous/demagnetized or nonhomogeneous/magnetized phase-space structures, separated by an out-of-equilibrium phase transition that depends on the initial energy u0 and magnetization M0 of the system. However, the magnetization also exhibits fluctuations around its mean value in time, which can provide additional insights into the nature of the QSS. In this study, the permutation entropy H and the statistical complexity C are used as tools to characterize the dynamical properties of these magnetization fluctuations. It is found that most data points lie above the entropy-complexity curve for stochastic processes with a power-law spectrum (k noise), suggesting that the magnetization retains more structure than purely stochastic processes. As the initial energy u0 increases, both H and C exhibit global minima that align closely with the critical energy u* separating magnetized and demagnetized QSSs. This agreement is particularly strong for M0 <= 0.4, where a first-order out-of-equilibrium phase transition has been reported for the HMF model. Below this transition, magnetized QSSs are associated with more ordered fluctuations, exhibiting fewer correlated structures. Above this transition, demagnetized QSSs are characterized by fluctuations that shift toward more complex and more disordered dynamics, with an increasing number of correlated structures as u0 increases.