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 ⟨Δx²⟩=2Dt^α(0<α≤2), then the thermal conductivity can be expressed in terms of the system size L as κ=cL^β with β=2-2/α. This result predicts that normal diffusion (α=1) implies normal heat conduction obeying the Fourier law (β=0) and that superdiffusion (α>1) implies anomalous heat conduction with a divergent thermal conductivity (β>0). More interestingly, subdiffusion (α<1) implies anomalous heat conduction with a convergent thermal conductivity (β<0), and, consequently, the system is a thermal insulator in the thermodynamic limit. Existing numerical data support our results.