We demonstrate that the condensed matter quantum systems encompassing two reservoirs connected by a junction permit a natural definition of flows of conserved measures, Renyi entropies. Such flows are similar to the flows of physical conserved quantities such as charge and energy. We develop a perturbation technique that permits efficient computation of Renyi entropy flows and analyze second- and fourth order contributions. Second-order approximation was shown to correspond directly to the transition events in the system and thereby to posess a set of "intuitive" features. The analysis of fourth-order corrections reveals a more complicated picture: the "intuitive" relations do not hold anymore, and the corrections exhibit divergencies in low-temperature limit manifesting an intriguing non-analytical dependence of the flows on coupling strength in the limit of weak couplings and vanishing temperatures.

10 pages