Rigorous Scaling Law for the Heat Current in Disordered Harmonic Chain
Résumé
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) \sim (T1−Tn)n^{−3/2}$ in the limit $n \rightarrow \infty$, 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.
Domaines
Physique mathématique [math-ph]
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