  We perform microscopic molecular dynamics simulations of particle chains with
an onsite anharmonicity to study relaxation of spatially homogeneous states to
equilibrium, and directly compare the simulations with the corresponding
Boltzmann-Peierls kinetic theory. The Wigner function serves as common
interface between the microscopic and kinetic level. We demonstrate
quantitative agreement after an initial transient time interval. In particular,
besides energy conservation, we observe the additional quasi-conservation of
the phonon density, defined via an ensemble average of the related microscopic
field variables and exactly conserved by the kinetic equations. On
super-kinetic time scales, density quasi-conservation is lost while energy
remains conserved, and we find evidence for eventual relaxation of the density
to its canonical ensemble value. However, the precise mechanism remains unknown
and is not captured by the Boltzmann-Peierls equations.
