  For low density gases the validity of the Boltzmann transport equation is
well established. The central object is the one-particle distribution function,
$f$, which in the Boltzmann-Grad limit satisfies the Boltzmann equation. Grad
and, much refined, Cercignani argue for the existence of this limit on the
basis of the BBGKY hierarchy for hard spheres. At least for a short kinetic
time span, the argument can be made mathematically precise following the
seminal work of Lanford. In this article a corresponding programme is
undertaken for weakly nonlinear, both discrete and continuum, wave equations.
Our working example is the harmonic lattice with a weakly nonquadratic on-site
potential. We argue that the role of the Boltzmann $f$-function is taken over
by the Wigner function, which is a very convenient device to filter the slow
degrees of freedom. The Wigner function, so to speak, labels locally the
covariances of dynamically almost stationary measures. One route to the phonon
Boltzmann equation is a Gaussian decoupling, which is based on the fact that
the purely harmonic dynamics has very good mixing properties. As a further
approach the expansion in terms of Feynman diagrams is outlined. Both methods
are extended to the quantized version of the weakly nonlinear wave equation.
  The resulting phonon Boltzmann equation has been hardly studied on a rigorous
level. As one novel contribution we establish that the spatially homogeneous
stationary solutions are precisely the thermal Wigner functions. For three
phonon processes such a result requires extra conditions on the dispersion law.
We also outline the reasoning leading to Fourier's law for heat conduction.
