Can disorder induce a finite thermal conductivity in 1D lattices?

We study heat conduction in one dimensional mass disordered harmonic and anharmonic lattices. It is found that the thermal conductivity $\kappa$ of the disordered anharmonic lattice is finite at low temperature, whereas it diverges as $\kappa \sim N^{0.43}$ at high temperature. Moreover, we demonstrate that a unique nonequilibrium stationary state in the disordered harmonic lattice does not exist at all.