  We study the dynamics of relaxation and thermalization in an exactly solvable
model of a particle interacting with a harmonic oscillator bath. Our goal is to
understand the effects of non-Markovian processes on the relaxational dynamics
and to compare the exact evolution of the distribution function with
approximate Markovian and Non-Markovian quantum kinetics. There are two
different cases that are studied in detail: i) a quasiparticle (resonance) when
the renormalized frequency of the particle is above the frequency threshold of
the bath and ii) a stable renormalized `particle' state below this threshold.
The time evolution of the occupation number for the particle is evaluated
exactly using different approaches that yield to complementary insights. The
exact solution allows us to investigate the concept of the formation time of a
quasiparticle and to study the difference between the relaxation of the
distribution of bare particles and that of quasiparticles. We derive a
non-Markovian quantum kinetic equation which resums the perturbative series and
includes off-shell effects. A Markovian approximation that includes off-shell
contributions and the usual Boltzmann equation (energy conserving) are obtained
from the quantum kinetic equation in the limit of wide separation of time
scales upon different coarse-graining assumptions. The relaxational dynamics
predicted by the non-Markovian, Markovian and Boltzmann approximations are
compared to the exact result. The Boltzmann approach is seen to fail in the
case of wide resonances and when threshold and renormalization effects are
important.
