Non-standard anomalous heat conduction in harmonic chains with correlated isotopic disorder

We address the general problem of heat conduction in one dimensional harmonic chain, with correlated isotopic disorder, attached at its ends to white noise or oscillator heat baths. When the low wavelength $\mu$ behavior of the power spectrum $W$ (of the fluctuations of the random masses around their common mean value) scales as $W(\mu)\sim \mu^\beta$, the asymptotic thermal conductivity $\kappa$ scales with the system size $N$ as $\kappa \sim N^{(1+\beta)/(2+\beta)}$ for free boundary conditions, whereas for fixed boundary conditions $\kappa \sim N^{(\beta-1)/(2+\beta)}$; where $\beta>-1$, which is the usual power law scaling for one dimensional systems. Nevertheless, if $W$ does not scale as a power law in the low wavelength limit, the thermal conductivity may not scale in its usual form $\kappa\sim N^{\alpha}$, where the value of $\alpha$ depends on the particular one dimensional model. As an example of the latter statement, if $W(\mu)\sim \exp(-1/\mu)/\mu^2$, $\kappa \sim N/(\log N)^3$ for fixed boundary conditions and $\kappa \sim N/\log(N)$ for free boundary conditions, which represent non-standard scalings of the thermal conductivity.