In physics, the Tsallis entropy is a generalization of the standard Boltzmann–Gibbs entropy. It is proportional to the expectation of the q-logarithm of a distribution.

The concept 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, introduced in 1967 within information theory.

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

where ${\displaystyle q}$ is a real parameter sometimes called entropic-index and ${\displaystyle k}$ a positive constant.

In the limit as ${\displaystyle q\to 1}$, the usual Boltzmann–Gibbs entropy is recovered, namely

where one identifies ${\displaystyle k}$ with the Boltzmann constant ${\displaystyle k_{B}}$.

For continuous probability distributions, we define the entropy as

where ${\displaystyle p(x)}$ is a probability density function.