  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.
