  We review recent rigorous mathematical results about the macroscopic
behaviour of harmonic chains with the dynamics perturbed by a random exchange
of velocities between nearest neighbor particles. The random exchange models
the effects of nonlinearities of anharmonic chains and the resulting dynamics
have similar macroscopic behaviour. In particular there is a superdiffusion of
energy for unpinned acoustic chains. The corresponding evolution of the
temperature profile is governed by a fractional heat equation. In non-acoustic
chains we have normal diffusivity, even if momentum is conserved.
