Scaling limits in Kinetic theory

Summer School - Lyon - 2019

**Talk by Jani Lukkarinen** (University of Helsinki)

*Kinetic theory of phonons and lattice waves*

We consider kinetic theory of weakly interacting waves which are arising from discrete wave equations such as eigenmodes of crystalline lattices. For random initial data in which field values have a spatially homogeneous distribution with sufficiently rapid decay of correlations, the evolution of second moments of the field may be approximated by solving the related Boltzmann-Peierls kinetic equation. Although the kinetic theory of waves and phonons has a long history and many uses in physics, the detailed relation between the original dynamical system and the kinetic equation, including a mathematically rigorous derivation, is still an open question. We discuss numerical and mathematical evidence which support the validity of this approximation, as well as its known limitations and modifications.