  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 $\gamma$.
