Fractional super-diffusion of energy in open harmonic chain with heat baths at the boundaries

We prove the hydrodynamic limit for a one-dimensional harmonic chain of $n+1$-interacting atoms with a random exchange of the momentum between neighbors. The system is open: at the left and right boundaries {are} attached to Langevin heat baths at temperatures $T_L$ and $T_R$, respectively. Under the super-diffusive scaling of space-time $(n u,n^{3/2}t)$, $u\in[0,1]$, $t\ge0$ we prove that the empirical profile of energy converges, as $n\to+\infty$, to the solution $T(t,u)$ of a fractional diffusion equation with {boundary conditons} determined by the temperatures of the heat baths. i.e. $T(t,0)=T_L$ and $T(t,1)=T_R$.