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

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