Non-extensive self-consistent thermodynamical theory

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In experimental physics, researchers have proposed Non-extensive self-consistent thermodynamic theory to describe phenomena observed in the Large Hadron Collider (LHC). This theory investigates a fireball for high-energy particle collisions, while using Tsallis non-extensive thermodynamics. [1] Fireballs lead to the bootstrap idea, or self-consistency principle, just as in the Boltzmann statistics used by Rolf Hagedorn.[2] Assuming the distribution function gets variations, due to possible symmetrical change, Abdel Nasser Tawfik applied the non-extensive concepts of high-energy particle production. [3] [4]

The motivation to use the non-extensive statistics from Tsallis[5] comes from the results obtained `by Bediaga et al.[6] They showed that with the substitution of the Boltzmann factor in Hagedorn's theory by the q-exponential function, it was possible to recover good agreement between calculation and experiment, even at energies as high as those achieved at the LHC, with q>1.

Non-extensive entropy for ideal quantum gas[edit]

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 S_q=S_q^{FD}+S_q^{BE} where S_q^{FD} is the non-extended version of the Fermi–Dirac entropy and 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.

Non-extensive partition function for ideal quantum gas[edit]

Using the entropy defined above, the partition function results are

 \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[edit]

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[1]

 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}}


\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 Hagedorn's theory.

Main results[edit]

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

 Z_q(V_o,T) \rightarrow \bigg(\frac{1}{\beta - \beta _o }\bigg)^{\alpha}


\alpha=\frac{\gamma V_o}{2\pi^2 \beta^{3/2}}\,,


 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 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.

Experimental evidence[edit]

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 I. Sena and A. Deppman.[5][6]


  1. ^ a b c A. Deppman, Physica A 391 (2012) 6380.
  2. ^ a b c R. Hagedorn, Suppl. Al Nuovo Cimento 3 (1965) 147.
  3. ^ a b c J. Cleymans and D. Worku, J. Phys. G: Nucl. Part. Phys. 39 (2012) 025006.
  4. ^ a b c J. Cleymans, G.I. Lykasov, A.S. Parvan, A.S. Sorin, O.V. Teryaev and D. Worku, arXiv:1302.1970 (2013).
  5. ^ I. Sena and A. Deppman, Eur. Phys. J. A 49 (2013) 17.
  6. ^ I. Sena and A. Deppman, AIP Conf. Proc. 1520, 172 (2013) - arXiv:1208.2952v1.