  Boltzmann equations are often used to describe the non-equilibrium
time-evolution of many-body systems in particle physics. Prominent examples are
the computation of the baryon asymmetry of the universe and the evolution of
the quark-gluon plasma after a relativistic heavy ion collision. However,
Boltzmann equations are only a classical approximation of the quantum
thermalization process, which is described by so-called Kadanoff-Baym
equations. This raises the question how reliable Boltzmann equations are as
approximations to the complete Kadanoff-Baym equations. Therefore, we present
in this article a detailed comparison of Boltzmann and Kadanoff-Baym equations
in the framework of a chirally invariant Yukawa-type quantum field theory
including fermions and scalars. The obtained numerical results reveal
significant differences between both types of equations. Apart from
quantitative differences, on a qualitative level the late-time universality
respected by Kadanoff-Baym equations is severely restricted in the case of
Boltzmann equations. Furthermore, Kadanoff-Baym equations strongly separate the
time scales between kinetic and chemical equilibration. In contrast to this
standard Boltzmann equations cannot describe the process of quantum-chemical
equilibration, and consequently also cannot feature the above separation of
time scales.
