Diffusion crossover from/to $q$-statistics to/from Boltzmann-Gibbs statistics in the classical inertial $\alpha$-XY ferromagnet

Abstract

We study the angular diffusion in a classical $d-$dimensional inertial XY model with interactions decaying with the distance between spins as $r^{-\alpha }$, wiht $\alpha ⩾0$. After a very short-time ballistic regime, with ${\sigma }_{\theta }^2\sim t^2$, a super-diffusive regime, for which ${\sigma }_{\theta }^2\sim t^{{\alpha }_D}$, with ${\alpha }_D\simeq 1\text{.}45$ is observed, whose duration covers an initial quasistationary state and its transition to a second plateau characterized by the Boltzmann-Gibbs temperature $T_{\text{BG}}$. Long after $T_{\text{BG}}$ is reached, a crossover to normal diffusion, ${\sigma }_{\theta }^2\sim t$, is observed. We relate, for the first time, via the expression ${\alpha }_D=2/(3-q)$, the anomalous diffusion exponent ${\alpha }_D$ 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, $f_q(x)\propto e_q(-\beta x^2)$, where $e_q(u)\equiv [1+(1-q)u{]}^{\frac{1}{1-q}}$ ($e_1(u)=\exp (u)$). For fixed size $N$ and large enough times, the index $q_{\theta }$ 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.