Rigorous scaling law for the heat current in disordered harmonic chain

Details Rigorous scaling law for the heat current in disordered harmonic chain Rigorous scaling law for the heat current in disordered harmonic chain

2010-03-04

arxiv.org/abs/1003.1076v1

Physics

Mathematical Physics

38 pages

Scientific paper

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.

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