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

Submitted to PRL on 18 Dec. 2002 and accepted for publication on 13 June 2003. The results were reported on the International Workshop and Seminar "Microscopic Chaos and Transport in Many-Particle Systems", at the MPIPKS Dresden from August 5 to August 25, 2002

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