Stability analysis of generalized entropies and q-exponential distributions

The q-exponential distributions, which are generalizations of the Zipf–Mandelbrot power-law distribution, are frequently observed in complex systems at their nonequilibrium stationary states persisting for long periods of time. It is therefore natural to imagine that these distributions may optimize a generalized entropy functional of a certain kind. In fact, there exist three such entropies known in the literature: the Rényi, Tsallis and normalized Tsallis entropies. Here it is shown that, among the three, only the Tsallis entropy is stable and can give an entropic basis for the q-exponential distributions. In particular, the Rényi entropy as well as the normalized Tsallis entropy cannot play any roles for generalizing statistical mechanics.