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The Hamiltonian Mean-Field (HMF) model belongs to a broad class ofstatistical physics models with non-additive Hamiltonians that reveal manynon-trivial properties, such as non-equivalence of statistical ensembles,ergodicity breaking, and negative specific heat. With this paper, we add tothis set another intriguing feature, which is that of super-diffusiveequilibrium dynamics. Using molecular dynamics techniques, we compare thediffusive properties of the HMF model in the quasi-stationary metastable state(QSS) and in the Boltzmann-Gibbs (BG) regime. In contrast to the current stateof knowledge, we show that L\'evy walks underlying super-diffusion in QSS donot disappear when the system settles in the thermodynamic equilibrium. Wedemonstrate that it is extremely difficult to distinguish QSS from the BGregime, by only examining the statistics of L\'evy walks in HMF particletrajectories. We construct a simple stochastic model based on the truncatedL\'{e}vy walks with rests that quantitatively resembles diffusion behaviorobserved in both stages of the HMF dynamics.

Truncated L\'{e}vy Walks and Superdiffusion in Boltzmann-Gibbs Equilibrium of the Hamiltonian Mean-Field Model