Phonon Boltzmann equations and kinetic scaling limits of lattice systems

We consider weakly perturbed discrete wave-equations in a kinetic scaling limit: space and time are both scaled by a square of the coupling constant which is then taken to zero. As shown in [J. Lukkarinen and H. Spohn, Arch. Ration. Mech. Anal. 183 (2007) 93-162], in three dimensions and with a random perturbation of the masses of the particles, the kinetic limit of the disorder-averaged Wigner transform of the solution satisfies a certain linear phonon Boltzmann equation. We also briefly discuss how a similar scheme can be applied for non-linear perturbations with suitable random initial data. As the resulting Boltzmann equations are irreversible, this provides an example how non-Hamiltonian energy transport can arise in such scaling limits of Hamiltonian systems.