Tsallis entropy

In physics, the Tsallis entropy is a generalization of the standard Boltzmann-Gibbs entropy. It was an extension put forward by Constantino Tsallis in 1988. It is defined as

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or in the discrete case

In this case, p denotes the probability distribution of interest, and q is a real parameter. In the limit as q → 1, the normal Boltzmann-Gibbs entropy is recovered.

The parameter q is a measure of the non-extensitivity of the system of interest. There are continuous and discrete versions of this entropic measure.

Various relationships

The discrete Tsallis entropy satisfies

where D_q is the q-derivative.

Non-extensivity

Given two independent systems A and B, for which the joint probability density satisfies

the Tsallis entropy of this system satisfies

From this result, it is evident that the parameter q is a measure of the departure from extensivity. In the limit when q = 1,

which is what is expected for an extensive system.

See also

• Rényi entropy