Tsallis entropy

In physics, the Tsallis entropy is a generalization of the standard Boltzmann–Gibbs entropy. It was introduced in 1988 by Constantino Tsallis as a basis for generalizing the standard statistical mechanics, and is identical in form to Havrda–Charvát structural α-entropy within Information Theory. In the scientific literature, the physical relevance of the Tsallis entropy has been debated. However, from the years 2000 on, an increasingly wide spectrum of natural, artificial and social complex systems have been identified which confirm the predictions and consequences that are derived from this nonadditive entropy, such as nonextensive statistical mechanics, which generalizes the Boltzmann–Gibbs theory.

Among the various experimental verifications and applications presently available in the literature, the following ones deserve a special mention:

Among the various available theoretical results which clarify the physical conditions under which Tsallis entropy and associated statistics apply, the following ones can be selected:

For further details a bibliography is available at http://tsallis.cat.cbpf.br/biblio.htm

Given a discrete set of probabilities ${\displaystyle \{p_{i}\}}$ with the condition ${\displaystyle \sum _{i}p_{i}=1}$, and ${\displaystyle q}$ any real number, the Tsallis entropy is defined as

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