Octave Boudouard

Non-extensive self-consistent thermodynamics

The non-extensive self-consistent thermodynamical theory keeps the concept of fireball for high-energy particle collisions at the same time that uses the Tsallis non-extensive thermodynamics. Fireballs lead naturally to the bootstrap-idea, or self-consistency principle, just as it happens in the case of the Boltzmann statistics used by Hagedorn. Assuming that the distribution function gets modifications due to possible symmetrical change, Abdel Nasser Tawfik applied the non-extensive concepts on high-energy particle production.

The motivation to use the non-extensive statistics from Tsallis comes from the results obtained by Bediaga et al., who showed that with the substitution of the Boltzmann factor which appears in the Hagedorn's theory by the q-exponential function, it was possible to recover the good agreement between calculation and experiment, even at energies as high as those achieved at LHC, with $q>1$.

Non-extensive entropy for ideal quantum gas

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

As shown by Conroy, Miller and Plastino and also by Cleymans and Worku, the entropy just defined leads to occupation numbers formulaes that reduces to the one used by Bediaga et al. and by C. Beck, and shows the power-like tails present in the distributions found in High Energy Physics experiments.

Non-extensive partition function for ideal quantum gas

Using the entropy defined above, the partition function results to be

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

Since experiments have shown that $q>1$, this restriction is adopted.

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

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

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

Self-consistency principle

The 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

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

in the limit of $m,E$ sufficiently large.

The self-consistency can be asymptotically achieved by choosing

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

and

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

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

Main results

With the mass spectrum and density of states given above, the asymptotic form of the partition function results to be

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

where

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

with

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

One immediate consequence of the expression for the partition function is the existence of a limiting temperature $T_{o}=1/\beta _{o}$. This result is equivalent to the important result obtained by Hagedorn. With these results it is expected that at sufficiently high energy collisions the fireball will present not only a constant temperature, but also constant entropic factor.

Experimental evidences

Experimental evidences of the existence of a limiting temperature and of a limiting entropic index can be found in the papers by J. Cleymans and collaborators, by Isaac Sena and A. Deppman.

References

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