From normal diffusion to superdiffusion of energy in the evanescent flip noise limit

We consider a harmonic chain perturbed by an energy conserving noise depending on a parameter $\gamma$. When $\gamma$ is of order one, the energy diffuses according to the standard heat equation after a space-time diffusive scaling. On the other hand, when $\gamma =0$, the energy superdiffuses according to a 3/4 fractional heat equation after a subdiffusive space-time scaling. In this paper, we study the existence of a crossover between these two regimes as a function of $\gamma$.