Tsallis entropy

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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

[S_q(p) = \left( 1 - \int p^q(x)\, dx \right),]

or in the discrete case

[S_q(p) = \left( 1 - \sum_x p^q(x) \right).]

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.

Non-extensivity

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

[p(A, B) = p(A) p(B),]

the Tsallis entropy of this system satisfies

[S_q(A,B) = S_q(A) + S_q(B) + (1-q)S_q(A) S_q(B).]

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

[S(A,B) = S(A) + S(B)]

which is what is expected for an extensive system.

Tsallis entropy

Non-extensivity

See also