  In this contribution we prove that the entropy of an N-body isolated system
can not decrease and the entropy production should be non-negative provided the
system possesses an equilibrium state. We define the entropy as a functional of
the set of n-particle reduced density operators (n=0,....,N) generalizing the
von Neumann fine-grained entropy formula. Additionally, as a consequence of our
analysis we find the expression of the equilibrium n-particle reduced density
operators which enter the definition of the entropy as well as the dissipated
energy in an irreversible process.
