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 $⟨\Delta x^2⟩=2D{t}^{\alpha }(0<\alpha \le 2)$, then the thermal conductivity can be expressed in terms of the system size $L$ as $\kappa =c{L}^{\beta }$ 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.