Tsallis entropy

In physics, the Tsallis entropy is a generalization of the standard Boltzmann-Gibbs entropy. In the scientific literature, the physical relevance of the Tsallis entropy is highly debated. It is not clear if any system obeys, and if so in which regime, the statistical mechanics that can be derived from such an approach.

The Tsallis entropy is defined as

$S_q(p) = {1 \over q - 1} \left( 1 - \int (p(x))^q\, dx \right),$

or in the discrete case

$S_q(p) = {1 \over q - 1} \left( 1 - \sum_x (p(x))^q \right),$

where S denotes entropy, p 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.

The Tsallis Entropy has been used along with the Principle of maximum entropy to derive the Tsallis distribution.

Various relationships

The discrete Tsallis entropy satisfies

$S_q = -\lim_{x\rightarrow 1}D_q \sum_i p_i^x$

where D_q is the q-derivative with respect to x. This may be compared to the standard entropy formula:

$S = -\lim_{x\rightarrow 1}\frac{d}{dx} \sum_i p_i^x$

Non-additivity

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 additivity. In the limit when q = 1,

$S(A,B) = S(A) + S(B),\,$

which is what is expected for an additive system. This property is sometimes referred to as "pseudo-additivity".

Exponential families

Many common distributions like the normal distribution belongs to the statistical exponential families. Tsallis entropy for an exponential family can be written (Nielsen, Nock, 2011) as

$H^T_q(p_F(x;\theta)) = \frac{1}{1-q} \left((e^{F(q\theta)-q F(\theta)}) E_p[e^{(q-1)k(x)}]-1 \right)$

where F is log-normalizer and k the term indicating the carrier measure. For multivariate normal, term k is zero, and therefore the Tsallis entropy is in closed-form.

See also

• Rényi entropy
• Tsallis distribution

References

• Nielsen, Frank; Nock, Richard (2011). "On Rényi and Tsallis entropies and divergences for exponential families". arXiv:1105.3259. 

• Frank Nielsen and Richard Nock (2012). "A closed-form expression for the Sharma-Mittal entropy of exponential families". Journal of Physics A: Mathematical and Theoretical. http://iopscience.iop.org/1751-8121/45/3/032003/. 

External links

• Tsallis Statistics, Statistical Mechanics for Non-extensive Systems and Long-Range Interactions