Rigorous Scaling Law for the Heat Current in Disordered Harmonic Chain

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 E J _n be the steady-state energy current across the chain, averaged over the masses. We prove that E J _n ~ (T ₁ − T _n )n ^−3/2 in the limit n → ∞, 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.