Anomalous heat conduction and anomalous diffusion in one dimensional systems

TL;DR

This paper establishes a theoretical link between anomalous diffusion types and heat conduction behavior in one-dimensional systems, predicting how different diffusion regimes affect thermal conductivity scaling.

Contribution

It provides a unified analytical framework connecting anomalous diffusion exponents to heat conduction laws in 1D systems, supported by numerical data.

Findings

Normal diffusion implies Fourier's law with finite conductivity

Superdiffusion leads to divergent thermal conductivity

Subdiffusion results in finite thermal conductivity and insulation

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 $<\Delta x^2> =2Dt^{\alpha} (0<\alpha\le 2)$, 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 results.