Tsallis distribution

The Tsallis distribution, also kown as a q-Gaussian, is a probability distribution arising from the optimisation of the Tsallis entropy. It is essentially a simple reparametrization of the Student's t-distribution introduced by W. Gosset in 1908 to describe small sample statistics. In Gosset's original presentation the degrees of freedom parameter $\nu$ was constrained to be a positive integer related to the sample size, but it is readily observed that Gosset's density function is valid for all real values of $\nu$. The reparametrization introduces the alternative parameter $q$ related to $\nu$ by

$q={\frac {(\nu +3)}{(\nu +1)}}$

with inverse

$\nu ={\frac {(3-q)}{(q-1)}}$

It is sometimes argued that the distribution is a generalization of the Student to negative and or non-integer degrees of freedom. However, almost the theory of the Student extends trivially to all real degrees of freedom, where the support of the distribution is now compact rather than infinite in the case of $\nu <0$. Furthermore, the case $\nu <0$ has to be covered by the two non-intersecting regions $q<1$ and $q>3$, so the original Student parametrisation is in many respects cleaner.

Applications

Economics (econophysics)

The Tsallis distribution has been used to describe the distribution of wealth (assets) between individuals .

See also

Internal links

• Constantino Tsallis
• Tsallis statistics
• Tsallis entropy

External links

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