Hydrodynamic limit for a disordered harmonic chain

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.

20 pages

References in corpus (3)

• Energy transport in stochastically perturbed lattice dynamics
• Scaling limits for gradient systems in random environment
• Ballistic and superdiffusive scales in macroscopic evolution of a chain of oscillators

Cited by in corpus (10)

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• Entropy growth during free expansion of an ideal gas
• Localization effects due to a random magnetic field on heat transport in a harmonic chain
• Equilibrium fluctuation for an anharmonic chain with boundary conditions in the Euler scaling limit
• Super diffusive length dependent thermal conductivity in one-dimensional materials with structural defects: longitudinal to transverse phonon scattering leads to $\kappa \propto L^{1/3}$ law
• Thermalization of isolated harmonic networks under conservative noise
• Spectral gap in mean-field $\mathcal{O}(n)$-model
• Equilibrium perturbations for stochastic interacting systems
• Large-deviations approach to thermalization: the case of harmonic chains with conservative noise
• Derivation of Euler equations from quantum and classical microscopic dynamics