Diffusion crossover between q statistics and Boltzmann-Gibbs statistics in the classical inertial α-XY ferromagnet

We study the angular diffusion in a classical d-dimensional inertial XY model with interactions decaying with the distance between spins as r^{-α}, with α⩾0. After a very short-time ballistic regime, with σ_{θ}^{2}∼t^{2}, a superdiffusive regime, for which σ_{θ}^{2}∼t^{α_{D}}, with α_{D}≃1.45 is observed, whose duration covers an initial quasistationary state and its transition to a second plateau characterized by the Boltzmann-Gibbs temperature T_{BG}. Long after T_{BG} is reached, a crossover to normal diffusion, σ_{θ}^{2}∼t, is observed. We relate via the expression α_{D}=2/(3-q), the anomalous diffusion exponent α_{D} with the entropic index q characterizing the time-averaged angles and momenta probability distribution functions (pdfs), which are given by the so called q-Gaussian distributions, f_{q}(x)∝e_{q}(-βx^{2}), where e_{q}(u)≡[1+(1-q)u]^{1/1-q} (e_{1}(u)=exp(u)). For fixed size N and large enough times, the index q_{θ} characterizing the angles pdf approaches unity, thus indicating a final relaxation to Boltzmann-Gibbs equilibrium. For fixed time and large enough N, the crossover occurs in the opposite sense.