Based on Tsallis entropy and the corresponding deformed exponential function, generalized distribution functions for bosons and fermions have been used since a while. However, aiming at a non-extensive quantum statistics further requirements arise fr
om the symmetric handling of particles and holes (excitations above and below the Fermi level). Naive replacements of the exponential function or cut and paste solutions fail to satisfy this symmetry and to be smooth at the Fermi level at the same time. We solve this problem by a general ansatz dividing the deformed exponential to odd and even terms and demonstrate that how earlier suggestions, like the kappa- and q-exponential behave in this respect.
Finite heat reservoir capacity and temperature fluctuations lead to modification of the well known canonical exponential weight factor. Requiring that the corrections least depend on the one-particle energy, we derive a deformed entropy, K(S). The re
sulting formula contains the Boltzmann-Gibbs, the Renyi and the Tsallis formulas as particular cases. For extreme large fluctuations (compared to the Gaussian case) a new, parameter-free entropy - probability relation emerges. This formula and the corresponding canonical equilibrium distribution are nearly Boltzmannian for high probability, but deviate from the classical result for low probability. In the extreme large fluctuation limit the canonical distribution resembles for low probability the cumulative Gompertz distribution.
LHC ALICE data are interpreted in terms of statistical power-law tailed pT spectra. As explanation we derive such statistical distributions for particular particle number fluctuation patterns in a finite heat bath exactly, and for general thermodynam
ical systems in the subleading canonical expansion approximately. Our general result, $q = 1 - 1/C + Delta T^2 / T^2$, demonstrates how the heat capacity and the temperature fluctuation effects compete, and cancel only in the standard Gaussian approximation.
