Tsallis statistics

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The term Tsallis statistics usually refers to the collection of mathematical functions and associated probability distributions that were originated by Constantino Tsallis. Using these tools, 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 and Lévy distributions can be created. This parameter q represents the degree of non-extensivity of the distribution. Tsallis statistics are useful for characterising complex, anomalous diffusion. Although the functions tend to their classical counterpart when q tends to 1, Tsallis' functions have nothing to do with q-analogs in the usual sense of the word.

Tsallis functions[edit]

The q-deformed exponential and logarithmic functions where first introduced in Tsallis statistics in 1994 [1]

q-exponential[edit]

The q-exponential is a deformation of the exponential function using the real parameter q.[2]

e_q(x) = \begin{cases}
\exp(x) & \text{if }q=1, \\[6pt]
[1+(1-q)x]^{1/(1-q)} & \text{if }q \ne 1 \text{ and } 1+(1-q)x >0, \\[6pt]
0^{1/(1-q)} & \text{if }q \ne 1\text{ and }1+(1-q)x \le 0, \\[6pt]
\end{cases}

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

q-logarithm[edit]

The q-logarithm is the inverse of q-exponential and a deformation of the logarithm using the real parameter q.[2]

\ln_q(x) = \begin{cases}
\ln(x) & \text{if }x\ge 0 \text{ and }q=1\\[8pt]
\dfrac{x^{1-q} - 1}{1-q} & \text{if }x\ge 0 \text{ and }q\ne 1\\[8pt]
\text{Undefined } & \text{if }x\le 0\\[8pt]
\end{cases}

These functions have the property that

\begin{cases}
e_q( \ln_q(x)) = x & (x>0)\\
\ln_q( e_q(x) ) = x & (0<e_q(x)<\infty)\\
\end{cases}

See also[edit]

References[edit]

  1. ^ Tsallis, Constantino (1994). "What are the numbers that experiments provide?". Quimica Nova 17: 468. 
  2. ^ a b Umarov, Sabir; Tsallis, Constantino and Steinberg, Stanly (2008). "On a q-Central Limit Theorem Consistent with Nonextensive Statistical Mechanics". Milan j. math. (Birkhauser Verlag) 76: 307–328. doi:10.1007/s00032-008-0087-y. Retrieved 2011-07-27. 
  • S. Abe, A.K. Rajagopal (2003). Letters, Science (11 April 2003), Vol. 300, issue 5617, 249–251. doi:10.1126/science.300.5617.249d
  • S. Abe, Y. Okamoto, Eds. (2001) Nonextensive Statistical Mechanics and its Applications. Springer-Verlag. ISBN 978-3-540-41208-3
  • G. Kaniadakis, M. Lissia, A. Rapisarda, Eds. (2002) "Special Issue on Nonextensive Thermodynamics and Physical Applications." Physica A 305, 1/2.

External links[edit]