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 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 (tε−1,xε−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 (tε−3/2, xε−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.

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. Nonlinearity, 2016, 29 (3), ⟨10.1088/0951-7715/29/3/962⟩. ⟨hal-01165878v2⟩