Non-extensive self-consistent thermodynamical theory

The starting point of the theory is entropy for a non-extensive quantum gas of bosons and fermions, as proposed by Conroy, Miller and Plastino,[1] which is given by ${\displaystyle S_{q}=S_{q}^{FD}+S_{q}^{BE}}$ where ${\displaystyle S_{q}^{FD}}$ is the non-extended version of the Fermi–Dirac entropy and ${\displaystyle S_{q}^{BE}}$ is the non-extended version of the Bose–Einstein entropy.

That group[2] and also Clemens and Worku,[3] the entropy just defined leads to occupation number formulas that reduce to Bediaga's. C. Beck,[4] shows the power-like tails present in the distributions found in high energy physics experiments.

Using the entropy defined above, the partition function results are

${\displaystyle \ln[1+Z_{q}(V_{o},T)]={\frac {V_{o}}{2\pi ^{2}}}\sum _{n=1}^{\infty }{\frac {1}{n}}\int _{0}^{\infty }dm\int _{0}^{\infty }dp\,p^{2}\rho (n;m)[1+(q-1)\beta {\sqrt {p^{2}+m^{2}}}]^{-{\frac {nq}{(q-1)}}}\,.}$

Since experiments have shown that ${\displaystyle q>1}$, this restriction is adopted.

Another way to write the non-extensive partition function for a fireball is

${\displaystyle Z_{q}(V_{o},T)=\int _{0}^{\infty }\sigma (E)[1+(q-1)\beta E]^{-{\frac {q}{(q-1)}}}dE\,,}$

where ${\displaystyle \sigma (E)}$ is the density of states of the fireballs.

Self-consistency implies that both forms of partition functions must be asymptotically equivalent and that the mass spectrum and the density of states must be related to each other by

${\displaystyle log[\rho (m)]=log[\sigma (E)]}$,

in the limit of ${\displaystyle m,E}$ sufficiently large.

The self-consistency can be asymptotically achieved by choosing[1]

${\displaystyle m^{3/2}\rho (m)={\frac {\gamma }{m}}{\big [}1+(q_{o}-1)\beta _{o}m{\big ]}^{\frac {1}{q_{o}-1}}={\frac {\gamma }{m}}[1+(q'_{o}-1)m]^{\frac {\beta _{o}}{q'_{o}-1}}}$

and

${\displaystyle \sigma (E)=bE^{a}{\big [}1+(q'_{o}-1)E{\big ]}^{\frac {\beta _{o}}{q'_{o}-1}}\,,}$

where ${\displaystyle \gamma }$ is a constant and ${\displaystyle q'_{o}-1=\beta _{o}(q_{o}-1)}$. Here, ${\displaystyle a,b,\gamma }$ are arbitrary constants. For ${\displaystyle q'\rightarrow 1}$ the two expressions above approach the corresponding expressions in Hagedorn's theory.

With the mass spectrum and density of states given above, the asymptotic form of the partition function is

${\displaystyle Z_{q}(V_{o},T)\rightarrow {\bigg (}{\frac {1}{\beta -\beta _{o}}}{\bigg )}^{\alpha }}$

where

${\displaystyle \alpha ={\frac {\gamma V_{o}}{2\pi ^{2}\beta ^{3/2}}}\,,}$

with

${\displaystyle a+1=\alpha ={\frac {\gamma V_{o}}{2\pi ^{2}\beta ^{3/2}}}\,.}$

One immediate consequence of the expression for the partition function is the existence of a limiting temperature ${\displaystyle T_{o}=1/\beta _{o}}$. This result is equivalent to Hagedorn's result.[2] With these results, it is expected that at sufficiently high energy, the fireball presents a constant temperature and constant entropic factor.

The connection between Hagedorn's theory and Tsallis statistics has been established through the concept of thermofractals, where it is shown that non extensivity can emerge from a fractal structure. This result is interesting because Hagedorn's definition of fireball characterizes it as a fractal.

Experimental evidence of the existence of a limiting temperature and of a limiting entropic index can be found in J. Cleymans and collaborators,[3][4] and by I. Sena and A. Deppman.[7][8]

• Self-consistency principle in high energy physics