  The quantum time evolution of \phi^4-field theory for a spatially homogeneous
system in 2+1 space-time dimensions is investigated numerically for
out-of-equilibrium initial conditions on the basis of the Kadanoff-Baym
equations including the tadpole and sunset self-energies. Whereas the tadpole
self-energy yields a dynamical mass, the sunset self-energy is responsible for
dissipation and an equilibration of the system. In particular we address the
dynamics of the spectral (`off-shell') distributions of the excited quantum
modes and the different phases in the approach to equilibrium described by
Kubo-Martin-Schwinger relations for thermal equilibrium states. The
investigation explicitly demonstrates that the only translation invariant
solutions representing the stationary fixed points of the coupled equation of
motions are those of full thermal equilibrium. They agree with those extracted
from the time integration of the Kadanoff-Baym equations in the long time
limit. Furthermore, a detailed comparison of the full quantum dynamics to more
approximate and simple schemes like that of a standard kinetic (on-shell)
Boltzmann equation is performed. Our analysis shows that the consistent
inclusion of the dynamical spectral function has a significant impact on
relaxation phenomena. The different time scales, that are involved in the
dynamical quantum evolution towards a complete thermalized state, are discussed
in detail. We find that far off-shell 1 <-> 3 processes are responsible for
chemical equilibration, which is missed in the Boltzmann limit. Finally, we
address briefly the case of (bare) massless fields. For sufficiently large
couplings $\lambda$ we observe the onset of Bose condensation, where our scheme
within symmetric \phi^4-theory breaks down.
