Rigorous scaling law for the heat current in disordered harmonic chain

Abstract

We study the energy current in a model of heat conduction, first considered in detail by Casher and Lebowitz. The model consists of a one-dimensional disordered harmonic chain of n i.i.d. random masses, connected to their nearest neighbors via identical springs, and coupled at the boundaries to Langevin heat baths, with respective temperatures T_1 and T_n. Let EJ_n be the steady-state energy current across the chain, averaged over the masses. We prove that EJ_n \sim (T_1 - T_n)n^{-3/2} in the limit n \to \infty, as has been conjectured by various authors over the time. The proof relies on a new explicit representation for the elements of the product of associated transfer matrices.

38 pages

References in corpus (2)

• Heat Transport in low-dimensional systems
• Gaussian estimates for spatially inhomogeneous random walks on ${\mathbf{Z}}^d$

Cited by in corpus (10)

• Steady states and universal conductance in a quenched Luttinger model
• Heat conduction in disordered harmonic lattices with energy conserving noise
• Quantum harmonic oscillator systems with disorder
• Drastic fall-off of the thermal conductivity for disordered lattices in the limit of weak anharmonic interactions
• Hydrodynamic limit for a disordered harmonic chain
• Localization effects due to a random magnetic field on heat transport in a harmonic chain
• Spectral gap in mean-field $\mathcal{O}(n)$-model
• Aspects of the disordered harmonic chain
• Derivation of Euler equations from quantum and classical microscopic dynamics
• Gaussian fluctuations of products of random matrices distributed close to the identity