  This paper is devoted to the derivation of a macroscopic fractional diffusion
equation describing heat transport in an anharmonic chain. More precisely, we
study here the so-called FPU-$\beta$ chain, which is a very simple model for a
one-dimensional crystal in which atoms are coupled to their nearest neighbors
by a harmonic potential, weakly perturbed by a quartic potential. The starting
point of our mathematical analysis is a kinetic equation: Lattice vibrations,
responsible for heat transport, are modeled by an interacting gas of phonons
whose evolution is described by the Boltzmann Phonon Equation. Our main result
is the rigorous derivation of an anomalous diffusion equation starting from the
linearized Boltzmann Phonon Equation
