A Universal Relation between Heat Conduction and Diffusion

We rigorously prove a useful equality relating the heat current autocorrelation function and the mean square displacement of the energy diffusion. From this equality, we are able to recover the existing theories for normal diffusion and normal heat conduction. Moreover, we are also able to obtain a connection between anomalous diffusion and anomalous heat conduction. We are results are applicable to all isotropic systems with any dimension.