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.

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

Olla, Stefano
2016

Abstract

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.
hydrodynamic limits, Euler equations, superdiffusion, fractional heat equation
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Utilizza questo identificativo per citare o creare un link a questo documento: http://hdl.handle.net/20.500.12571/19313
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