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

TL;DR

This paper investigates the macroscopic evolution of a one-dimensional chain of harmonic oscillators with conserved energy, momentum, and volume, revealing hyperbolic and superdiffusive scales where different energy components evolve according to Euler and fractional heat equations.

Contribution

It demonstrates the existence of distinct space-time scales for energy evolution, showing hyperbolic limits follow Euler equations and thermal energy exhibits superdiffusive behavior described by fractional heat equations.

Findings

Conserved quantities follow Euler system at hyperbolic scale.

Thermal energy remains stationary at hyperbolic scale.

Thermal energy evolves via fractional heat equation at superdiffusive scale.

Abstract

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 (t$\epsilon$--1,x$\epsilon$--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 (t$\epsilon$--3/2, x$\epsilon$--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.