The term Tsallis statistics usually refers to the collection of mathematical functions and associated probability distributions that were originated by Constantino Tsallis. Using these tools, it is possible to derive Tsallis distributions from the optimization of the Tsallis entropic form. A continuous real parameter q can be used to adjust the distributions so that distributions which have properties intermediate to that of Gaussian and Lévy distributions can be created. This parameter q represents the degree of non-extensivity of the distribution. Tsallis statistics are useful for characterising complex, anomalous diffusion. Although the functions tend to their classical counterpart when q tends to 1, Tsallis' functions have nothing to do with q-analogs in the usual sense of the word.
The q-deformed exponential and logarithmic functions where first introduced in Tsallis statistics in 1994 
Note that the q-exponential in Tsallis statistics is different from a version used elsewhere.
These functions have the property that
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