Heat Flow in Regular and Disordered Harmonic Chains

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Abstract
We investigate the steady state heat flux J in a large harmonic crystal containing different masses whose ends are in contact with heat baths at different temperatures. Calling ΔT the temperature difference and L the distance between the ends, we are interested in the behavior of J/ΔT as L→∞. For a perfectly periodic harmonic crystal, J/ΔT approaches a fixed positive value as L→∞ corresponding to an infinite heat conductivity. We show that this will be true also for a general one-dimensional harmonic chain (arbitrary distribution of different masses) if the spectral measure of the infinite chain contains an absolutely continuous part. We also show that for an infinite chain containing two different masses, the cumulative frequency distribution is continuous and that the spectrum is not exhausted by a denumerable number of points, i.e., the spectrum cannot consist entirely of point eigenvalues with a denumerable number of limit points. Using a theorem of Matsuda and Ishii, we show that for a random chain, corresponding to the mass at each site being an independent random variable, the heat flux approaches zero as L→∞, with probability one. This implies that the spectrum of a disordered chain has, with probability one, no absolutely continuous part.