Ballistic and superdiffusive scales in macroscopic evolution of a chain of oscillators

We consider a one dimensional infinite acoustic chain of harmonic oscillators whose dynamics is perturbed by a random exchange of velocities, such that the energy and momentum of the chain are conserved. Consequently, the evolution of the system has only three conserved quantities: volume, momentum and energy. We show the existence of two space--time scales on which the en- ergy of the system evolves. On the hyperbolic scale (t$\epsilon$--1,x$\epsilon$--1) the limits of the conserved quantities satisfy a Euler system of equa- tions, while the thermal part of the energy macroscopic profile re- mains stationary. Thermal energy starts evolving at a longer time scale, corresponding to the superdiffusive scaling (t$\epsilon$--3/2, x$\epsilon$--1) and follows a fractional heat equation. We also prove the diffusive scal- ing limit of the Riemann invariants - the so called normal modes, corresponding to the linear hyperbolic propagation.