Rigorous scaling law for the heat current in disordered harmonic chain

TL;DR

This paper rigorously establishes the asymptotic behavior of heat current in a disordered harmonic chain, confirming a conjectured scaling law and providing a new mathematical representation for transfer matrices.

Contribution

The authors prove a precise scaling law for heat current in a disordered harmonic chain, advancing theoretical understanding of heat conduction in disordered systems.

Findings

Heat current scales as n^{-3/2} with chain length n.

The proof introduces a new explicit transfer matrix representation.

The result confirms a longstanding conjecture in the field.

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.