Tsallis statistics

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Tsallis statistics is a family of mathematical tools and probability distributions created by Constantino Tsallis. They come from maximizing a generalized entropy called Tsallis entropy. By changing a real parameter q, you can shape distributions that sit between the normal (Gaussian) and heavy-tailed (Lévy) types. The parameter q reflects how non-additive or non-extensive the system is. These ideas are useful for describing complex systems and anomalous diffusion.

A key part of Tsallis statistics is the q-deformed exponential and logarithm, introduced in 1994. The q-exponential is a deformation of the ordinary exponential, and the q-logarithm is its inverse. When q = 1, these reduce to the standard exponential and logarithm. The limit can be understood by noting that as q approaches 1 (more precisely, with N = 1/(1−q) going to infinity), (1 + x/N)^N approaches e^x and N(x^(1/N) − 1) approaches ln x. The q-logarithm is also related to the Box–Cox transformation for the case q = 1 − λ, a concept introduced by Box and Cox in 1964.