  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).
