Anomalous Heat Diffusion

TL;DR

This paper establishes a rigorous relation between anomalous energy spread and heat flux autocorrelation in solids, revealing how anomalous diffusion influences thermal conductivity scaling.

Contribution

It derives a fundamental relation linking MSD and heat flux autocorrelation for anomalous heat diffusion, providing a new theoretical framework for understanding thermal transport.

Findings

Derived a second-order differential relation between MSD and heat flux autocorrelation.

Established a time-local Helfand-moment relation for energy diffusion.

Linked anomalous energy diffusion to scaling behavior of thermal conductivity.

Abstract

Consider anomalous energy spread in solid phases, i.e., $MSD= \int (x -{\langle x \rangle}_E)^2 \rho_E(x,t)dx \propto t^{\beta}$, as induced by a small initial excess energy perturbation distribution $\rho_{E}(x,t=0)$ away from equilibrium. The associated total thermal equilibrium heat flux autocorrelation function $C_{JJ}(t)$ is shown to obey rigorously the intriguing relation, $d^2 MSD/dt^2 = 2C_{JJ}(t)/(k_BT^2c)$, where $c$ is the specific volumetric heat capacity. Its integral assumes a time-local Helfand-moment relation; i.e. $ dMSD/dt|_{t=t_s} = 2/(k_BT^2c)\int_0^{t_s} C_{JJ}(s)ds$, where the chosen cut-off time $t_s$ is determined by the maximal signal velocity for heat transfer. Given the premise that the averaged nonequilibrium heat flux is governed by an anomalous heat conductivity, energy diffusion scaling determines a corresponding anomalous thermal conductivity scaling behaviour.