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

:$S\_q(p)\; =\; \{1\; over\; q\; -\; 1\}\; left(\; 1\; -\; int\; p^q(x),\; dx\; ight),$

or in the discrete case

:$S\_q(p)\; =\; \{1\; over\; q\; -\; 1\}\; left(\; 1\; -\; sum\_x\; p^q(x)\; ight).$

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

:$S\_q\; =\; -\; left\; [\; D\_q\; sum\_i\; p\_i^x\; ight\; ]\; \_\{x=1\}$

where "D"_"q" is the q-derivative.

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 $|1-q|$ 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.

See also

*Rényi entropy

External links

* [http://www.cscs.umich.edu/~crshalizi/notabene/tsallis.html Tsallis Statistics, Statistical Mechanics for Non-extensive Systems and Long-Range Interactions]