On the Markov evolution of the $\rho$-matrix of a subsystem

Evolution of the reduced density matrix for a subsystem is studied to determine deviations from its Markov character for a system consisting of a closed chain of $N$ oscillators with one of them serving as a subsystem. The dependence on $N$ and on the coupling of the two subsystems is investigated numerically. The found deviations strongly depend on $N$ and the coupling. In the most beneficial case with $N-1=100$ and the coupling randomized in its structure the deviations fall with the evolution time up 3\%. In other cases they remain to be of the order 30\% or even more.