  A generalization of the Gibbs-von Neumann relative entropy is proposed based
on the quantum BBGKY [Bogolyubov-Born-Green-Kirkwood-Yvon] hierarchy as the
nonequilibrium entropy for an N-body system. By using a generalization of the
Liouville-von Neumann equation describing the evolution of a density super-
operator, it is demonstrated that the entropy production for an isolated system
is non-negative, which provides an arrow of time. Moreover, following the
procedure of non-equilibrium thermodynamics a master matrix is introduced for
which a mi- croscopic expression is obtained. Then, the quantum Boltzmann
equation is derived in terms of a transition superoperator related to that
master matrix.
