Anomalous Heat Diffusion

Consider anomalous energy spread in solid phases, i.e., <Δx(2)(t)>E≡∫(x-<x>E)(2)ρE(x,t)dx∝t(β), as induced by a small initial excess energy perturbation distribution ρE(x,t=0) away from equilibrium. The second derivative of this variance of the nonequilibrium excess energy distribution is shown to rigorously obey the intriguing relation d(2)<Δx(2)(t)>E/dt2=2CJJ(t)/(kBT(2)c), where CJJ(t) equals the thermal equilibrium total heat flux autocorrelation function and c is the specific volumetric heat capacity. Its integral assumes a time-local Helfand-like relation. Given that the averaged nonequilibrium heat flux is governed by an anomalous heat conductivity, the energy diffusion scaling determines a corresponding anomalous thermal conductivity scaling behavior.