Anomalous heat conduction and anomalous diffusion in one-dimensional systems

We establish a connection between anomalous heat conduction and anomalous diffusion in one-dimensional systems. It is shown that if the mean square of the displacement of the particle is <Deltax(2)>=2Dt(alpha)(0<alphaless than or equal to2), then the thermal conductivity can be expressed in terms of the system size L as kappa=cL(beta) with beta=2-2/alpha. This result predicts that normal diffusion (alpha=1) implies normal heat conduction obeying the Fourier law (beta=0) and that superdiffusion (alpha>1) implies anomalous heat conduction with a divergent thermal conductivity (beta>0). More interestingly, subdiffusion (alpha<1) implies anomalous heat conduction with a convergent thermal conductivity (beta<0), and, consequently, the system is a thermal insulator in the thermodynamic limit. Existing numerical data support our results.