Consider anomalous energy spread in solid phases, i.e., MSD=∫(x−⟨x⟩E)2ρE(x,t)dx∝tβ, as induced by a
small initial excess energy perturbation distribution ρE(x,t=0) away
from equilibrium. The associated total thermal equilibrium heat flux
autocorrelation function CJJ(t) is shown to obey rigorously the intriguing
relation, d2MSD/dt2=2CJJ(t)/(kBT2c), where c is the specific
volumetric heat capacity. Its integral assumes a time-local Helfand-moment
relation; i.e. dMSD/dt∣t=ts=2/(kBT2c)∫0tsCJJ(s)ds, where
the chosen cut-off time ts 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