A connection was established between anomalous heat conduction and anomalous diffusion in 1-dimensional systems. It was shown that if the mean square of the displacement of the particle was equal to 2Dtα (where α = 0 to 2), then the thermal conductivity could be expressed in terms of the system size L as κ = cLβ; with β = 2–2/α. This result predicts that normal diffusion (α = 1) implied normal heat conduction obeying the Fourier law (β = 0) and that super-diffusion (α > 1) implied anomalous heat conduction with a divergent thermal conductivity (β > 0). Moreover, sub-diffusion (α < 1) implied anomalous heat conduction with a convergent thermal conductivity (β < 0), and the system was therefore a thermal insulator in the thermodynamic limit. Existing numerical data support these results.

Anomalous Heat Conduction and Anomalous Diffusion in One-Dimensional Systems. B.Li, J.Wang: Physical Review Letters, 2003, 91[4], 044301 (4pp)