From anomalous energy diffusion to Levy walks and heat conductivity in   one-dimensional systems
  The evolution of infinitesimal, localized perturbations is investigated in a one-dimensional diatomic gas of hard-point particles (HPG) and thereby connected to energy diffusion. As a result, a Levy walk description, which was so far invoked to explain anomalous heat conductivity in the context of non-interacting particles is here shown to extend to the general case of truly many-body systems. Our approach does not only provide a firm evidence that energy diffusion is anomalous in the HPG, but proves definitely superior to direct methods for estimating the divergence rate of heat conductivity which turns out to be $0.333\pm 0.004$, in perfect agreement with the dynamical renormalization--group prediction (1/3). 
P. Cipriani, S. Denisov, and A. Politi