BALLISTIC AND SUPERDIFFUSIVE SCALES IN MACROSCOPIC EVOLUTION OF A CHAIN OF OSCILLATORS

Tomasz Komorowski, Stefano Olla
unpublished
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 energy of the system evolves. On the hyperbolic scale (tǫ −1 , xǫ −1) the limits of the conserved quantities satisfy a Euler
more » ... tem of equations , while the thermal part of the energy macroscopic profile remains stationary. Thermal energy starts evolving at a longer time scale, corresponding to the superdiffusive scaling (tǫ −3/2 , xǫ −1) and follows a fractional heat equation. We also prove the diffusive scaling limit of the Riemann invariants-the so called normal modes, corresponding to the linear hyperbolic propagation.
fatcat:qadv4tmgkfghrib3itmu7avgu4