A family of fractional diffusion equations derived from stochastic harmonic chains with long-range interactions

We consider one-dimensional infinite chains of harmonic oscillators with stochastic perturbations and long-range interactions which have polynomial decay rate $|x|^{-\theta}, x \to \infty, \theta > 1$, where $x \in \mathbb{Z}$ is the interaction range. We prove that if $2< \theta \le 3$, then the time evolution of the macroscopic thermal energy distribution is superdiffusive and governed by a fractional diffusion equation with exponent $\frac{3}{7-\theta}$, while if $\theta > 3$, then the exponent is $\frac{3}{4}$. The threshold is $\theta = 3$ because the derivative of the dispersion relation diverges as $k \to 0$ when $\theta \le 3$.