Anomalous Heat Diffusion

Consider anomalous energy spread in solid phases, i.e., (E) = integral(x - < x >(E))(2)rho(E)(x, t)dx proportional to t(beta), as induced by a small initial excess energy perturbation distribution rho(E)(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)(E)/dt(2) = 2C(JJ)(t)/(k(B)T(2)c), where C-JJ(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.