Heat Conduction in the Disordered Harmonic Chain Revisited

Abstract

A general formulation is developed to study heat conduction in disordered harmonic chains with arbitrary heat baths, satisfying the fluctuation-dissipation theorem. A simple formal expression for the heat current $J$ is obtained, from which its asymptotic system-size $\left(N\right)$ dependence is extracted. It is shown that “thermal conductivity” depends not just on the system itself but also on the spectral properties of the heat baths. As special cases we recover earlier results that gave $J\sim {1/N}^{3/2}$ for fixed boundaries and $J\sim {1/N}^{1/2}$ for free boundaries. Other choices give other power laws including the “Fourier behavior” $J\sim 1/N$.