  More and more works deal with statistical systems far from equilibrium,
dominated by unidirectional stochastic processes augmented by rare resets. We
analyze the construction of the entropic distance measure appropriate for such
dynamics. We demonstrate that a power-like nonlinearity in the state
probability in the master equation naturally leads to the Tsallis
(Havrda-Charv\'at, Acz\'el-Dar\'oczy) q-entropy formula in the context of
seeking for the maximal entropy state at stationarity. A few possible
applications of a certain simple and linear master equation to phenomena
studied in statistical physics are listed at the end.
