  Boltzmann equations are often used to study the thermal evolution of particle
reaction networks. Prominent examples are the computation of the baryon
asymmetry of the universe and the evolution of the quark-gluon plasma after
relativistic heavy ion collisions. However, Boltzmann equations are only a
classical approximation of the quantum thermalization process which is
described by the so-called Kadanoff-Baym equations. This raises the question
how reliable Boltzmann equations are as approximations to the full
Kadanoff-Baym equations. Therefore, we present in this paper a detailed
comparison between the Kadanoff-Baym and Boltzmann equations in the framework
of a scalar Phi^4 quantum field theory in 3+1 space-time dimensions. The
obtained numerical solutions reveal significant discrepancies in the results
predicted by both types of equations. Apart from quantitative discrepancies, on
a qualitative level the universality respected by the Kadanoff-Baym equations
is severely restricted in the case of Boltzmann equations. Furthermore, the
Kadanoff-Baym equations strongly separate the time scales between kinetic and
chemical equilibration. This separation of time scales is absent for the
Boltzmann equation.
