A solid crystal is modeled as a chain of oscillators (=the atoms) coupled to their nearest neighbors by an harmomic potential perturbed by an anharmonic potential. The goal of this talk is to derive a macroscopic equation (the heat equation) describing the propagation of heat in the chain. As an intermediary step between the discrete chain model and the macroscopic equation, we will use the Boltzmann phonon equation to describe the evolution of vibrations through the lattice. For the particular model that we are studying (called the FPU-beta chain in the literature) we will prove that heat diffusion is correctly described by a fractional diffusion equation rather than the standard heat equation.
