  The thermalization process is studied in an expanding parton gas using the
Boltzmann equation with two types of collision terms. In the relaxation time
approximation we determine the criteria under which a time-dependent relaxation
time leads to thermalization of the partons. We calculate the entropy
production due to collisions for the general time-dependent relaxation time. In
a perturbative QCD approach on the other hand, we can estimate the parton
collision time and its dependence on expansion time. The effective `out of
equilibrium' collision time differs from the standard transport relaxation
time, $\tau_{\rm tr}\simeq(\alpha_s^2\ln(1/\alpha_s)T)^{-1}$, by a weak time
dependence. It is in both cases Debye screening and Landau damping that
regulate the singular forward scattering processes. We find that the parton gas
does thermalize eventually but only after having undergone a phase of free
streaming and gradual equilibration where considerable entropy is produced
(``after-burning"). The final entropy and thus particle density depends on the
collision time as well as the initial conditions (a ``memory effect"). Results
for entropy production are presented based upon various model estimates of
early parton production.
