Anomalous heat conduction and anomalous diffusion in one dimensional systems

Abstract

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 $<Δx^2> =2Dt^α (0<α\le 2)$, 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 a normal diffusion ($α=1$) implies a normal heat conduction obeying the Fourier law ($β=0$), a superdiffusion ($α>1$) implies an anomalous heat conduction with a divergent thermal conductivity ($β>0$), and more interestingly, a subdiffusion ($α<1$) implies an anomalous heat conduction with a convergent thermal conductivity ($β<0$), consequently, the system is a thermal insulator in the thermodynamic limit. Existing numerical data support our results.