The term Tsallis statistics usually refers to the collection of mathematical functions and associated probability distributions that were originated by Constantino Tsallis. Using that collection, it is possible to derive Tsallis distributions from the optimization of the Tsallis entropic form. A continuous real parameter ''q'' can be used to adjust the distributions, so that distributions which have properties intermediate to that of

Gaussian Carl Friedrich Gauss (1777–1855) is the eponym of all of the topics listed below. There are over 100 topics all named after this German mathematician and scientist, all in the fields of mathematics, physics, and astronomy. The English eponymo ...

and

Lévy distribution In probability theory and statistics, the Lévy distribution, named after Paul Lévy, is a continuous probability distribution for a non-negative random variable. In spectroscopy, this distribution, with frequency as the dependent variable, is k ...

s can be created. The parameter ''q'' represents the degree of non- extensivity of the distribution. Tsallis statistics are useful for characterising complex,

anomalous diffusion Anomalous diffusion is a diffusion process with a non-linear relationship between the mean squared displacement (MSD), \langle r^(\tau )\rangle , and time. This behavior is in stark contrast to Brownian motion, the typical diffusion process descr ...

Tsallis functions

The ''q''-deformed exponential and logarithmic functions were first introduced in Tsallis statistics in 1994. However, the ''q''-logarithm is the

Box–Cox transformation In statistics, a power transform is a family of functions applied to create a monotonic transformation of data using power functions. It is a data transformation technique used to stabilize variance, make the data more normal distribution-like, i ...

for

q=1-\lambda

, proposed by

George Box George Edward Pelham Box (18 October 1919 – 28 March 2013) was a British statistician, who worked in the areas of quality control, time-series analysis, design of experiments, and Bayesian inference. He has been called "one of the gre ...

and David Cox in 1964.

''q''-exponential

The ''q''-exponential is a deformation of the exponential function using the real parameter ''q''. :

\end

Note that the ''q''-exponential in Tsallis statistics is different from a version used elsewhere.

''q''-logarithm

The ''q''-logarithm is the inverse of ''q''-exponential and a deformation of the

logarithm In mathematics, the logarithm of a number is the exponent by which another fixed value, the base, must be raised to produce that number. For example, the logarithm of to base is , because is to the rd power: . More generally, if , the ...

using the real parameter ''q''. :

\end

Inverses

These functions have the property that Function of inverse

\begin e_q( \ln_q(x)) = x & (x>0)\\

\ln_q( e_q(x) ) = x & (0

Analysis

The

q\to 1

limits of the above expression can be understood by considering

\left(1+\frac\right)^N \approx ^x

for the exponential function and

N\left(x^-1\right)\approx \log(x)

for the logarithm, in the limit

N = \frac\to \infty

References

* S. Abe, A.K. Rajagopal (2003). Letters, ''Science'' (11 April 2003), Vol. 300, issue 5617, 249–251. * S. Abe, Y. Okamoto, Eds. (2001) ''Nonextensive Statistical Mechanics and its Applications.'' Springer-Verlag. * G. Kaniadakis, M. Lissia, A. Rapisarda, Eds. (2002) "Special Issue on Nonextensive Thermodynamics and Physical Applications." ''Physica'' A 305, 1/2.

External links

Tsallis statistics on arxiv.org
{{Tsallis Statistical mechanics