Tsallis statistics

The q-exponential is a deformation of the exponential function using the real parameter q.[3]

${\displaystyle e_{q}(x)={\begin{cases}\exp(x)&{\text{if }}q=1,\\[6pt][1+(1-q)x]^{1/(1-q)}&{\text{if }}q\neq 1{\text{ and }}1+(1-q)x>0,\\[6pt]0^{1/(1-q)}&{\text{if }}q\neq 1{\text{ and }}1+(1-q)x\leq 0,\\[6pt]\end{cases}}}$

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

The q-logarithm is the inverse of q-exponential and a deformation of the logarithm using the real parameter q.[3]

${\displaystyle \ln _{q}(x)={\begin{cases}\ln(x)&{\text{if }}x>0{\text{ and }}q=1\\[8pt]{\dfrac {x^{1-q}-1}{1-q}}&{\text{if }}x>0{\text{ and }}q\neq 1\\[8pt]{\text{Undefined }}&{\text{if }}x\leq 0\\[8pt]\end{cases}}}$

These functions have the property that

${\displaystyle {\begin{cases}e_{q}(\ln _{q}(x))=x&(x>0)\\\ln _{q}(e_{q}(x))=x&(0<e_{q}(x)<\infty )\\\end{cases}}}$

The ${\displaystyle q\to 1}$ limits of the above expression can be understood by considering ${\displaystyle \left(1+{\frac {x}{N}}\right)^{N}\approx {\rm {e}}^{x}}$ for the exponential function and ${\displaystyle N\left(x^{\frac {1}{N}}-1\right)\approx \log(x)}$ for the logarithm.