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For low density gases the validity of the Boltzmann transport equation iswell established. The central object is the one-particle distribution function,$f$, which in the Boltzmann-Grad limit satisfies the Boltzmann equation. Gradand, much refined, Cercignani argue for the existence of this limit on thebasis of the BBGKY hierarchy for hard spheres. At least for a short kinetictime span, the argument can be made mathematically precise following theseminal work of Lanford. In this article a corresponding programme isundertaken for weakly nonlinear, both discrete and continuum, wave equations.Our working example is the harmonic lattice with a weakly nonquadratic on-sitepotential. We argue that the role of the Boltzmann $f$-function is taken overby the Wigner function, which is a very convenient device to filter the slowdegrees of freedom. The Wigner function, so to speak, labels locally thecovariances of dynamically almost stationary measures. One route to the phononBoltzmann equation is a Gaussian decoupling, which is based on the fact thatthe purely harmonic dynamics has very good mixing properties. As a furtherapproach the expansion in terms of Feynman diagrams is outlined. Both methodsare extended to the quantized version of the weakly nonlinear wave equation.  The resulting phonon Boltzmann equation has been hardly studied on a rigorouslevel. As one novel contribution we establish that the spatially homogeneousstationary solutions are precisely the thermal Wigner functions. For threephonon processes such a result requires extra conditions on the dispersion law.We also outline the reasoning leading to Fourier's law for heat conduction.Comment: special issue on "Kinetic Theory", Journal of Statistical Physics,  improved versio

The Phonon Boltzmann Equation, Properties and Link to Weakly Anharmonic Lattice Dynamics