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

We consider a one dimensional infinite acoustic chain of harmonic oscillators whose dynamics are 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 energy of the system evolves. On the hyperbolic scale $left(t{{epsilon}^{-1}},x{{epsilon}^{-1}}ight)$ the limits of the conserved quantities satisfy a Euler system of equations, while the thermal part of the macroscopic energy profile remains stationary. Thermal energy starts evolving at a longer time scale, corresponding to superdiffusive scaling $left(t{{epsilon}^{-3/2}},x{{epsilon}^{-1}}ight)$ , and follows a fractional heat equation. We also prove the diffusive scaling limit of the Riemann invariants—the so-called normal modes, corresponding to linear hyperbolic propagation.