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

Tomasz Komorowski (1) , Stefano Olla (2)

1 PAN - Institut of Mathematics - Polish Academy of Sciences
2 CEREMADE - CEntre de REcherches en MAthématiques de la DEcision

We consider a one dimensional infinite 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: mass, momentum and energy. We show the existence of two space–time scales on which the energy of the system evolves. On the hyperbolic scale the limits of the conserved quantities satisfy a Euler system of equations, while the thermal part of the energy macroscopic profile remains stationary. This part of energy starts evolving at a longer time scale, corresponding to the superdiffusive scaling 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.

Domaines: Mécanique statistique [cond-mat.stat-mech]; Physique mathématique [math-ph]; Probabilités [math.PR]

Tomasz Komorowski, Stefano Olla. BALLISTIC AND SUPERDIFFUSIVE SCALES IN MACROSCOPIC EVOLUTION OF A CHAIN OF OSCILLATORS. 2015. ⟨hal-01165878v1⟩