Hydrodynamic limit for a disordered harmonic chain

TL;DR

This paper proves that a disordered harmonic chain's macroscopic behavior converges to Euler equations under hyperbolic scaling, highlighting the role of scale separation over ergodicity and showing the temperature profile remains static.

Contribution

It demonstrates the hydrodynamic limit for a disordered harmonic chain, emphasizing scale separation as the key factor over ergodicity in deriving Euler equations.

Findings

Mechanical and thermal modes decouple due to Anderson localization.

Energy conservation holds out of thermal equilibrium.

Temperature profile remains unchanged over time.

Abstract

We consider a one-dimensional unpinned chain of harmonic oscillators with random masses. We prove that after hyperbolic scaling of space and time the distributions of the elongation, momentum and energy converge to the solution of the Euler equations. Anderson localization decouples the mechanical modes from the thermal modes, allowing the closure of the energy conservation equation even out of thermal equilibrium. This example shows that the derivation of Euler equations rests primarily on scales separation and not on ergodicity. Furthermore, it follows from our proof that the temperature profile does not evolve in any space-time scale.