  Consider anomalous energy spread in solid phases, i.e., $MSD= \int (x -{< x
>}_E)^2 \rho_E(x,t)dx \propto t^{\beta}$, 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
necessarily determines a corresponding anomalous thermal conductivity scaling
behavior.
