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.