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 a Markovian stochastic process. The transition rates between system states, which characterize the process, are determined by Fermi's golden rule. Such evolution is time-irreversible and leads to the equidistribution of probabilities across every state of the system. Furthermore, we investigate the time dependence of the mean numbers of particles in single-particle states and find that, under the given assumptions, it is governed by the Boltzmann collision integral. The proposed mechanism that transforms time-reversible unitary evolution into time-irreversible stochastic evolution is non-coherence. In the presented theory, the non-coherence is not associated with interaction with a heat bath, but rather with the finite spectral width of quantum states. This understanding of non-coherence is analogous to the one used in wave optics. Thus, we present a novel approach to the famous arrow of time problem.