Malnormal subgroup

2013-09-20T08:59:25Z

85.169.42.221: correcting a typo

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 distribution]]s from the optimization of the [[Tsallis entropy|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 [[normal distribution|Gaussian]] and [[Lévy distribution]]s 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-analog]]s in the usual sense of the word.

==Tsallis functions==
The q-deformed exponential and logarithmic functions where first introduced in Tsallis statistics in 1994 <ref name="Tsallis1994">{{cite journal |last1=Tsallis |first1=Constantino |year=1994 |title= What are the numbers that experiments provide? |journal=Quimica Nova |volume=17 |pages=468}}</ref>

===q-exponential===
The q-exponential is a deformation of the [[exponential]] function using the parameter ''q''.<ref name="Umarov2008">{{cite journal |last1=Umarov |first1=Sabir |coauthors=Tsallis, Constantino and Steinberg, Stanly |year=2008 |title=On a q-Central Limit Theorem Consistent with Nonextensive Statistical Mechanics |journal=Milan j. math. |volume=76 |issue= |pages=307–328 |publisher=Birkhauser Verlag |doi=10.1007/s00032-008-0087-y |url=http://www.cbpf.br/GrupPesq/StatisticalPhys/pdftheo/UmarovTsallisSteinberg2008.pdf |accessdate=2011-07-27}}</ref>{{clarify|date=June 2011|reason=need to specify valid range of q}}

:<math>e_q(x) = [1+(1-q)x]^{1 \over 1-q} \text{ for } 1+(1-q)x >0 \text{, otherwise } e_q(x) = 0 \text{ if } q<1 \text{, } =+\infty \text{ if }q>1</math>
:<math>e_q(x) = \exp(x) \text{ if } q = 1 </math>

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

===q-logarithm===
The q-logarithm is the inverse of q-exponential and a deformation of the [[logarithm]] using the parameter ''q''.<ref name="Umarov2008"/>

:<math>\ln_q(x) = {{x^{1-q} - 1} \over {1-q}} </math>
:<math>\ln_q(x) = \ln(x) \text{ if } q = 1 </math>

These functions have the property that
:<math>e_q( \ln_q(x)) = x </math>
and
:<math>\ln_q( e_q(x) ) = x .</math>

== See also ==
* [[Tsallis entropy]]
* [[Tsallis distribution]]
* [[q-Gaussian]]
* [[q-exponential distribution]]

==References==
<references/>
* 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==
*[http://xstructure.inr.ac.ru/x-bin/theme3.py?level=1&index1=183123 Tsallis statistics on arxiv.org]

{{Tsallis}}

[[Category:Statistical mechanics]]

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Malnormal subgroup