Nonextensive statistical mechanics: Applications to high energy physics

Nonextensive statistical mechanics was proposed in 1988 on the basis of the nonadditive entropy S q = k [1 − ∑ i p q i ]/(q − 1) (q ∈ R) which generalizes that of Boltzmann-Gibbs S BG = S 1 = −k ∑ i pi ln pi. This theory extends the applicability of standard statistical mechanics in order to also cover a wide class of anomalous systems which violate usual requirements such as ergodicity. Along the last two decades, a variety of applications have emerged in natural, artificial and social systems, including high energy phenomena. A brief review of the latter will be presented here, emphasizing some open issues.


Introduction
Standard statistical mechanics is based on the Boltzmann-Gibbs (BG) entropy S BG = −k W i=1 p i ln p i ( W i=1 p i = 1), where W is the number of microscopic configurations of the system.This extremely powerful theory -one of the pillars of contemporary physics -has exhibited very many successes along 140 years, in particular through its celebrated distribution for thermal equilibrium p i ∝ e −βE i , E i being the energy of the corresponding microstate.However, as any other human intellectual construct, it has a restricted domain of validity.For nonlinear dynamical manybody systems the usual requirement is ergodicity, which is guaranted by strong chaos (i.e., by a positive maximal Lyapunov exponent for classical systems).For nonergodic systems (typically for systems whose maximal Lyapunov exponent vanishes), which is quite frequently the case of the so-called complex systems, there is no general reason for legitimately using the BG theory.For (some of) such anomalous systems, a generalization of the BG theory has been proposed in 1988 [1].It is frequently referred to as nonextensive statistical mechanics [2-4] because the total energy of such systems typically is nonextensive, i.e., not proportional to the total number of elements of the system.This generalized theory is based on the entropy It can be straightforwardly verified that, if A and B are two probabilistically independent systems (i.e., if a e-mail: tsallis@cbpf.brwhich exhibits that, in contrast with S BG which is additive, the entropy S q is nonadditive for q 1.This nonadditivity will in fact enable it to be extensive (i.e., proportional to the number of elements of the system) for various classes of systems (see for instance [5,6]).

Connection to Thermodynamics
To generalize BG statistical mechanics for the canonical ensemble (from [7]), we optimize S q with the constraints and where is the so-called escort distribution [8].It follows that p i = P 1/q i W j=1 P 1/q j .There are various converging reasons for being appropriate to impose the energy constraint with the {P i } instead of with the original {p i }.The full discussion of this delicate point is beyond the present scope.However, some of these intertwined reasons are explored in [2].By imposing Eq. (4), we follow [7], which in turn reformulates the results presented in [1,9].The passage from one to the other of the various existing formulations of the above optimization problem are discussed in detail in [7,10].
The entropy optimization yields, for the stationary state, β being the Lagrange parameter associated with the constraint (4).Eq. ( 6) makes explicit that the probability distribution is, for fixed β q , invariant with regard to the arbitrary choice of the zero of energies.The stationary state (or (meta)equilibrium) distribution (6) can be rewritten as follows: with and The form (9) is particularly convenient for many applications where comparison with experimental or computational data is involved.Also, it makes clear that p i asymptotically decays like 1/E 1/(q−1) i for q > 1, and has a cutoff for q < 1, instead of the exponential decay with E i for q = 1.
The connection to thermodynamics is established in what follows.It can be proved that with T ≡ 1/(kβ).Also we prove, for the free energy, where ln q Z q = ln q Zq − βU q .
This relation takes into account the trivial fact that, in contrast with what is usually done in BG statistics, the energies {E i } are here referred to U q in (6).It can also be proved as well as relations such as In fact, the entire Legendre transformation structure of thermodynamics is q-invariant, which is both remarkable and welcome.

In diverse systems
The nonadditive entropy S q and its associated nonetensive statistical mechanics have been applied to a wide variety of natural, artificial and social systems.Among others we may mention (i) The velocity distribution of (cells of) Hydra viridissima follows a q = 3/2 probability distribution function (PDF) [11]; (ii) The velocity distribution of (cells of) Dictyostelium discoideum follows a q = 5/3 PDF in the vegetative state and a q = 2 PDF in the starved state  [46].The systematic study of metastable or long-living states in long-range versions of magnetic models such as the Ising [47] and Heisenberg [48] ones, or in hydrogen-like atoms [49][50][51] might provide further illustrations.

In high energy physics
Connections of nonextensive statistics with a specific area of solar physics, astrophysics, high energy physics, and related areas, were pioneered by Quarati and collaborators (see [52], among others), who advanced the possibility of this theory being useful in the discussion of the flux of solar neutrinos.A few years later, it was realized that the transverse momenta distribution of the hadronic jets resulting from electron-positron annihilation are well described by distributions associated with q-exponentials [53,54]: see Figs. 1 and 2. The energy distribution of cosmic rays has been satisfactorily fitted in [55,56] with distributions related to q-exponentials: see Fig. 3.The distributions of returns of magnetic field fluctuations in the solar wind plasma as observed in data from Voyager 1 [57] and from Voyager 2 [58] has provided the values associated with the so called q-triplet: see Figs. 4 e 5. Similar results have been obtained in the study of interstellar turbulence [59] (see Figs. 6 and 7), in X-ray-emitting binary systems [60] (see Fig. 8), and in the distribution of stellar rotational velocities in the Pleiades [61].
It is important to address here the fact that the distribution of transverse momenta in high-energy collisions of proton-proton, and heavy nuclei (e.g., Pb-Pb and Au-Au) have received and are receiving great attention [62][63][64][65][66][67][68]: see illustrative examples in Figs.9-15.Several such data have been summarized in [69]: see Fig. 16.We realize that for such collisions the typical values of q are usually close to 1.10, apparently never above say 1.20-1.25.It remains as a challenging problem to precisely understand why (Is it a hadronization of quark matter in a sort of metastable state before attaining ergodicity?).In any case, it was shown in [71] that QCD calculations and q-statistical calculations can be consistent for q ≃ 1.1: see Fig.     3.The particular case q = 1 corresponds to the Hagedorn 1965 theory.It is advanced in [54] the possibility that q approaches the value 11/9 in the E → ∞ limit.See details in [53].Comparison of QCD diffusion calculation with its corresponding within q-statistics: they are consistent for q = 1.11.See details in [71]. 17 .

Fig. 1 .
Fig. 1.Distributions of transverse momenta for four typical values of the collision energy.See details in [53].

Fig. 2 .
Fig.2.Dependence of the index q (a) and the temperature T 0 (b) on the collision energies of Fig.3.The particular case q = 1 corresponds to the Hagedorn 1965 theory.It is advanced in[54] the possibility that q approaches the value 11/9 in the E → ∞ limit.See details in[53].

Fig. 3 .
Fig.3.Flux of cosmic rays.Curiously enough, the upper value of the index q is very close to 11/9.See details in[55,56].

Fig. 4 .
Fig.4.The q-triplet as obtained from data of the Voyager 1. See details in[57].

Fig. 10 .
Fig.10.Dependence of the temperature T on the index q for production of negative pions in different reactions.See details in[62].

Fig. 12 .
Fig.12.The index q (top) and the temperature (bottom) extracted from hadronic spectra assuming quark coalescence at a sudden hadron formation.See details in[63].

Fig. 14 .
Fig. 14.Transverse momenta distributions of various hadrons in pp collisions, as measured by the PHENIX Collaboration, corresponding to 200 GeV.At this energy it has been obtained q ≃ 1.10.See details in [67].

Fig. 15 .
Fig. 15.Transverse momenta distributions of charged hadrons in pp and heavy ion collisions, as measured in Brookhaven.See details in [68].

Fig. 16 .
Fig. 16.Index q obtained at various energies.See details in [69].The black dots indicate recent CMS results.The red dot indicates the value obtained in [70].

Fig. 17.
Fig. 17.Comparison of QCD diffusion calculation with its corresponding within q-statistics: they are consistent for q = 1.11.See details in[71].
[12]; (iii) The velocity distribution in defect turbulence [13]; (iv) The velocity distribution of cold atoms in a dissipative optical lattice [14]; (v) The velocity distribution during silo drainage [15,16]; (vi) The velocity distribution in a drivendissipative 2D dusty plasma, with q = 1.08 ± 0.01 and q = 1.05 ± 0.01 at temperatures of 30000 K and 61000 K respectively [17]; (vii) The spatial (Monte Carlo) distributions of a trapped 136 Ba + ion cooled by various classical buffer gases at 300 K [18]; (viii) The distributions of price returns and stock volumes at the stock exchange, as well as the volatility smile [19-22]; (ix) Biological evolution [23]; (x) The distributions of returns in the Ehrenfest's dog-flea model [24,25]; (xi) The distributions of returns in the coherent noise model [26]; (xii) The distributions of returns of the avalanche sizes in the self-organized critical Olami-Feder-Christensen model, as well as in real earthquakes [27]; (xiii) The distributions of angles in the HMF model [28]; (xiv) Turbulence in electron plasma [29]; (xv) The relaxation in various paradigmatic spin-glass substances through neutron spin echo experiments [30]; (xvi) Various properties directly related with the time dependence of the width of the ozone layer around the Earth [31]; (xvii) Various properties for conservative and dissipative nonlinear dynamical systems [32-41]; (xviii) The degree distribution of (asymptotically) scale-free networks [42,43]; (xix) Tissue radiation response [44]; (xx) Overdamped motion of interacting particles [45]; (xxi) Rotational population in molecular spectra in plasmas