Heat conduction in harmonic chains with Levy-type disorder

We consider heat transport in one-dimensional harmonic chains attached at its ends to Langevin heat baths. The harmonic chain has mass impurities where the separation $d$ between any two successive impurities is randomly distributed according to a power-law distribution $P(d)\sim 1/d^{\alpha+1}$, being $\alpha>0$. In the regime where the first moment of the distribution is well defined ($1<\alpha<2$) the thermal conductivity $\kappa$ scales with the system size $N$ as $\kappa\sim N^{(\alpha-3)/\alpha}$ for fixed boundary conditions, whereas for free boundary conditions $\kappa\sim N^{(\alpha-1)/\alpha}$ if $N\gg1$. When $\alpha=2$, the inverse localization length $\lambda$ scales with the frequency $\omega$ as $\lambda\sim \omega^2 \ln \omega$ in the low frequency regime, due to the logarithmic correction, the size scaling law of the thermal conductivity acquires a non-closed form. When $\alpha>2$, the thermal conductivity scales as in the uncorrelated disorder case. The situation $\alpha<1$ is only analyzed numerically, where $\lambda(\omega)\sim \omega^{2-\alpha}$ which leads to the following asymptotic thermal conductivity: $\kappa \sim N^{-(\alpha+1)/(2-\alpha)}$ for fixed boundary conditions and $\kappa \sim N^{(1-\alpha)/(2-\alpha)}$ for free boundary conditions.