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

Abstract

The Hamiltonian Mean-Field (HMF) model belongs to a broad class of statistical physics models with non-additive Hamiltonians that reveal many non-trivial properties, such as non-equivalence of statistical ensembles, ergodicity breaking, and negative specific heat. With this paper, we add to this set another intriguing feature, which is that of super-diffusive equilibrium dynamics. Using molecular dynamics techniques, we compare the diffusive properties of the HMF model in the quasi-stationary metastable state (QSS) and in the Boltzmann-Gibbs (BG) regime. In contrast to the current state of knowledge, we show that L\'evy walks underlying super-diffusion in QSS do not disappear when the system settles in the thermodynamic equilibrium. We demonstrate that it is extremely difficult to distinguish QSS from the BG regime, by only examining the statistics of L\'evy walks in HMF particle trajectories. We construct a simple stochastic model based on the truncated L\'{e}vy walks with rests that quantitatively resembles diffusion behavior observed in both stages of the HMF dynamics.