Anomalous heat conduction and anomalous diffusion in nonlinear lattices, single walled nanotubes, and billiard gas channels

TL;DR

This paper establishes a universal relationship between anomalous diffusion and heat conduction in low-dimensional systems, linking diffusion exponents to thermal conductivity scaling, supported by numerical evidence.

Contribution

It introduces a theoretical framework connecting anomalous diffusion exponents with heat conduction behavior across various low-dimensional systems.

Findings

Normal diffusion implies Fourier law heat conduction.

Superdiffusion leads to divergent thermal conductivity.

Subdiffusion results in finite thermal conductivity, indicating insulators.

Abstract

We study anomalous heat conduction and anomalous diffusion in low dimensional systems ranging from nonlinear lattices, single walled carbon nanotubes, to billiard gas channels. We find that in all discussed systems, the anomalous heat conductivity can be connected with the anomalous diffusion, namely, if energy diffusion is $\sigma^2(t)\equiv <\Delta x^2> =2Dt^{\alpha} (0<\alpha\le 2)$, then the thermal conductivity can be expressed in terms of the system size $L$ as $\kappa = cL^{\beta}$ 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 theoretical prediction.