We review the idea of generating non-extensive stationary distributions based on abstract composition rules of the subsystem energies, in particular the parton cascade method, using a Boltzmann equation with relativistic kinematics and modified two-body energy composition rules. The thermodynamical behavior of such model systems is investigated. As an application hadronic spectra with power-law tails are analyzed in the framework of a quark coalescence model. PACS. 21.65.Qr quark matter -25.75.Ag global features in relativistic heavy ion collisions -05.20.Dd kinetic theory

The scaling of charged hadron fragmentation functions to the Tsallis distribution for the momentum fraction 0.01 x 0.2 is presented for various e + e − collision energies. A possible microcanonical generalisation of the Tsallis distribution is proposed, which gives good agreement with measured data up to x ≈ 1. The proposal is based on superstatistics and a Koba -Nielsen -Olesen (KNO) like scaling of multiplicity distributions in e + e − experiments.

Abstract:The analysis of high-energy particle collisions is an excellent testbed for the non-extensive statistical approach. In these reactions we are far from the thermodynamical limit. In small colliding systems, such as electron-positron or nuclear collisions, the number of particles is several orders of magnitude smaller than the Avogadro number; therefore, finite-size and fluctuation effects strongly influence the final-state one-particle energy distributions. Due to the simple characterization, the description of the identified hadron spectra with the Boltzmann-Gibbs thermodynamical approach is insufficient. These spectra can be described very well with Tsallis-Pareto distributions instead, derived from non-extensive thermodynamics. Using the q-entropy formula, we interpret the microscopic physics in terms of the Tsallis q and T parameters. In this paper we give a view on these parameters, analyzing identified hadron spectra from recent years in a wide center-of-mass energy range. We demonstrate that the fitted Tsallis-parameters show dependency on the center-of-mass energy and particle species (mass). Our findings are described well by a QCD (Quantum Chromodynamics) inspired parton evolution ansatz. Based on this comprehensive study, apart from the evolution, both mesonic and baryonic components found to be non-extensive (q > 1), besides the mass ordered hierarchy observed in the parameter T. We also study and compare in details the theory-obtained parameters for the case of PYTHIA8 Monte Carlo Generator, perturbative QCD and quark coalescence models.

Certain fluctuations in particle number, n, at fixed total energy, E, lead exactly to a cut-power law distribution in the one-particle energy, ω, via the induced fluctuations in the phase-space volume ratio, Ω n (E − ω)/Ω n (E) = (1 − ω/E) n . The only parameters are 1/T = β = n /E and q = 1 − 1/ n + ∆n 2 / n 2 . For the binomial distribution of n one obtains q = 1−1/k, for the negative binomial q = 1+1/(k +1). These results also represent an approximation for general particle number distributions in the reservoir up to second order in the canonical expansion ω ≪ E. For general systems the average phase-space volume ratio e S(E−ω) /e S(E)to second order delivers q = 1 − 1/C + ∆β 2 / β 2 with β = S ′ (E) and C = dE/dT heat capacity. However, q = 1 leads to non-additivity of the Boltzmann-Gibbs entropy, S. We demonstrate that a deformed entropy, K(S), can be constructed and used for demanding additivity, i.e., q K = 1. This requirement leads to a second order differential equation for K(S). Finally, the generalized q-entropy formula, K(S) = p i K(− ln p i ), contains the Tsallis, Rényi and Boltzmann-Gibbs-Shannon expressions as particular cases. For diverging variance, ∆β 2 we obtain a novel entropy formula.