Non-coherent evolution of closed weakly interacting system leads to equidistribution of probabilities of microstates

We introduce a concept of non-coherent evolution of macroscopic quantum systems. We show that for weakly interacting systems such evolution is described by a symmetric continuous-time Markov chain. The non-diagonal elements of its transition rate matrix can be 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 values of occupation number of 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 for weakly interacting systems.