  We introduce a concept of non-coherent evolution of macroscopic quantum
systems. We show that for weakly interacting systems such evolution is a
Markovian stochastic process. The rates of transitions between states of the
system, that characterize the process, are found by Fermi's golden rule. Such
evolution is time-irreversible and leads to the equidistribution of
probabilities across every state of the system. Further, we search for time
dependence of mean numbers of particles in single-particle states and find that
under made assumptions it is governed by Boltzmann collision integral. In this
theory, the non-coherence is the mechanism that transforms time-reversible
unitary evolution into time-irreversible stochastic evolution. Thus we present
the possible solution for the famous time-arrow problem.
