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.
We’re sorry, something doesn't seem to be working properly.
Please try refreshing the page. If that doesn't work, please contact support so we can address the problem.
References
Aizenman M., Graf G.M.: Localization bounds for an electron gas. J. Phys. A Math. Gen. 31(32), 6783–6806 (1998)
Aizenman M., Molchanov S.: Localization at large disorder and at extreme energies: an elementary derivations. Commun. Math. Phys. 157(2), 245–278 (1993)
Ajanki O., Huveneers F.: Rigorous scaling law for the heat current in disordered harmonic chain. Commun. Math. Phys. 301(3), 841–883 (2011)
Anderson P.W.: Absence of diffusion in certain random lattices. Phys. Rev. 109, 1492 (1958)
Basile G., Olla S., Spohn H.: Energy transport in stochastically perturbed lattice dynamics. Arch. Ration. Mech. Anal. 195(1), 171–203 (2010)
Bernardin C.: Homogenization results for a linear dynamics in random Glauber type environment. Annales de l’I.H.P. Probabilités Et Statistiques 48(3), 792–818 (2012)
Bernardin, C., Olla, S.:Thermodynamics and non-equilibrium macroscopic dynamics of chains of anharmonic oscillators (in preparation)
Braxmeier-Even N., Olla S.: Hydrodynamic limit for a Hamiltonian system with boundary conditions and conservative noise. Arch. Ration. Mech. Anal. 213(2), 561–585 (2014)
Casher A., Lebowitz J.L.: Heat flow in regular and disordered harmonic chains. J. Math. Phys. 12(8), 1701–1711 (1971)
Dhar A.: Heat conduction in the disordered harmonic chain revisited. Phys. Rev. Lett. 86, 5882–5885 (2001)
Dobrushin R.L., Pellegrinotti A., Suhov Yu.M., Triolo L.: One-dimensional harmonic lattice caricature of hydrodynamics. J. Stat. Phys. 43(3-4), 571–607 (1986)
Gonçalves P., Jara M.: Scaling limits for gradient systems in random environment. J. Stat. Phys. 131(4), 691–716 (2008)
Halperin B.I.: Quantized Hall conductance, current-carrying edge states, and the existence of extended states in a two-dimensional disordered potential. Phys. Rev. B 25(4), 2185–2190 (1982)
Jara M., Landim C.: Quenched non-equilibrium central limit theorem for a tagged particle in the exclusion process with bond disorder. Annales de l’I.H.P. Probabilittés Et Statistiques 44(2), 341–361 (2008)
Jara M., Komorowski T., Olla S.: Superdiffusion of energy in a chain of harmonic oscillators with noise. Commun. Math. Phys. 339(2), 407–453 (2015)
Komorowski T., Olla S.: Ballistic and superdiffusive scales in the macroscopic evolution of a chain of oscillators. Nonlinearity 29(3), 962 (2016)
Kunz H., Souillard B.: Sur le spectre des opérateurs aux différences finies aléatoires. Commun. Math. Phys. 78(2), 201–246 (1980)
Olla S., Varadhan S.R.S., Yau H.T.: Hydrodynamic limit for a Hamiltonian system with weak noise. Commun. Math. Phys. 155(3), 523–560 (1993)
Rieder Z., Lebowitz J.L., Lieb E.: Properties of a harmonic crystal in a stationary nonequilibrium state. J. Math. Phys. 8, 1073 (1967)
Rubin R.J., Greer W.L.: Abnormal lattice thermal conductivity of a one-dimensional, harmonic isotopically disordered crystal. J. Math. Phys. 12, 1686–1701 (1971)
Verheggen T.: Transmission coefficient and heat conduction of a harmonic chain with random masses: asymptotic estimates on products of random matrices. Commun. Math. Phys. 68(3), 69–82 (1979)
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by H. Spohn
Rights and permissions
About this article
Cite this article
Bernardin, C., Huveneers, F. & Olla, S. Hydrodynamic Limit for a Disordered Harmonic Chain. Commun. Math. Phys. 365, 215–237 (2019). https://doi.org/10.1007/s00220-018-3251-4
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00220-018-3251-4