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-Charvat, Aczel-Daroczy) 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.
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