Power laws in elementary and heavy-ion collisions A story of fluctuations and nonextensivity?

We review from the point of view of nonextensive statistics the ubiquitous presence in elementary and heavy-ion collisions of power law distributions. Special emphasis is placed on the conjecture that this is just a reflection of some intrinsic fluctuations existing in the hadronic systems considered. These systems are summarily described by a single parameter q playing the role of a nonextensivity measure in the nonextensive statistical models based on Tsallis entropy. This paper is part of the Topical Issue Statistical Power Law Tails in High-Energy Phenomena.     

The thermal Green functions in nonextensive quantum statistical mechanics

The thermal Green functions of the quantum-mechanical harmonic oscillator are constructed within the framework of nonextensive statistical mechanics with normalized q -expectation values. For the Tsallis index q greater than unity, these functions are found to be expressed analytically in terms of the Hurwitz zeta function. It is found that influence of the nonextensivity on the time-ordered thermal propagator is relevant only at the “on-shell” states. In particular, the finite-temperature contribution to the thermal propagator becomes enhanced for the strong nonextensivity. Received 30 September 1998     

Generalized statistics framework for rate distortion theory

Variational principles for the rate distortion (RD) theory in lossy compression are formulated within the ambit of the generalized nonextensive statistics of Tsallis, for values of the nonextensivity parameter satisfying 0<q<1 and q>1. Alternating minimization numerical schemes to evaluate the nonextensive RD function, are derived. Numerical simulations demonstrate the efficacy of generalized statistics RD models.     

Consequences of temperature fluctuations in observables measured in high-energy collisions

We review the consequences of intrinsic, nonstatistical temperature fluctuations as seen in observables measured in high-energy collisions. We do this from the point of view of nonextensive statistics and Tsallis distributions. Particular attention is paid to multiplicity fluctuations as a first consequence of temperature fluctuations, to the equivalence of temperature and volume fluctuations, to the generalized thermodynamic fluctuations relations allowing us to compare fluctuations observed in different parts of the phase space, and to the problem of the relation between Tsallis entropy and Tsallis distributions. We also discuss the possible influence of conservation laws on these distributions and provide some examples of how one can get them without considering temperature fluctuations.     

Specific heat of the harmonic oscillator within generalized equilibrium statistics

Summary Within the generalized equilibrium statistics recently introduced by Tsallis (p _n ∝[1−β(q−-1) _εn ]^1/(q−)), we calculate the thermal dependence of the specific heat corresponding to a harmonic-oscillator-like spectrum, namely ε _n ω(n−α) (∀ω>0,n=0,1,2,...). The influences ofq and α are exhibited. Physically inaccessible and/or thermally frozen gaps are obtained in the low-temperature region, and, forq>1, oscillations are observed in the high-temperature region. The specific heat of the two-level system is also shown.     

Modified exponential function: the connection between nonextensivity and q-deformation

Tsallis thermostatistics has deep-rooted connection with quantum group formalism. Assuming that the modification of the standard exponential function considered in Tsallis thermostatistics has the same functional form as the one appearing in the q-calculus formalism and using the appropriate internal energy constraint, we derive the temperature dependent connection between the nonextensivity parameter and deformation parameter.     

Intermittency at critical transitions and aging dynamics at the onset of chaos

We recall that at both the intermittency transitions and the Feigenbaum attractor, in unimodal maps of non-linearity of order ζ > 1, the dynamics rigorously obeys the Tsallis statistics. We account for theq-indices and the generalized Lyapunov coefficients λ^q that characterize the universality classes of the pitchfork and tangent bifurcations. We identify the Mori singularities in the Lyapunov spectrum at the onset of chaos with the appearance of a special value for the entropic indexq. The physical area of the Tsallis statistics is further probed by considering the dynamics near criticality and glass formation in thermal systems. In both cases a close connection is made with states in unimodal maps with vanishing Lyapunov coefficients.     

A unified grand canonical description of the nonextensive thermostatistics of the quantum gases: Fractal and fractional approach

In this paper, the particles of quantum gases, that is, bosons and fermions are regarded as g-ons which obey fractional exclusion statistics. With this point of departure the thermostatistical relations concerning the Bose and Fermi systems are unified under the g-on formulation where a fractal approach is adopted. The fractal inspired entropy, the partition function, distribution function, the thermodynamics potential and the total number of g-ons have been found for a grand canonical g-on system. It is shown that from the g-on formulation; by a suitable choice of the parameters of the nonextensivity q, the parameter of the fractional exclusion statistics g, nonextensive Tsallis as well as extensive (q=1) standard thermostatistical relations of the Bose and Fermi systems are recovered. Received 17 September 1999     

Nonextensive quantum method for itinerant-electron ferromagnetism: Factorization approach

Magnetic and thermodynamical properties of itinerant-electron (metallic) ferromagnets described by the Hubbard model have been discussed with the use of the generalized Fermi-Dirac (GFD) distribution for nonextensive quantum systems. We have derived the GFD distribution within the superstatistics, which is equivalent to that obtained by the maximum-entropy method to the Tsallis entropy with the factorization approximation. By using the Hartree-Fock approximation to the electron-electron interaction in the Hubbard model, we have calculated magnetic moment, energy, specific heat and Curie-Weiss-type spin susceptibility, as functions of the temperature and entropic index q expressing the degree of the nonextensivity: q=1.0 corresponds to the Boltzmann-Gibbs statistics. It has been shown that by increasing the nonextensivity of |q−1|, the temperature dependence of magnetic moment becomes more significant and the low-temperature electronic specific heat is very much increased. This is attributed to enlarged Stoner excitations in the GFD distribution, which is elucidated by an analysis with the use of the generalized Sommerfeld expansion. We discuss the difference and similarity between the effects of the nonextensivity on metallic and insulating ferromagnets.     

The maximization of Tsallis entropy with complete deformed functions and the problem of constraints

By only requiring the q deformed logarithms (q exponentials) to possess arguments chosen from the entire set of positive real numbers (all real numbers), we show that the q-logarithm (q exponential) can be written in such a way that its argument varies between 0 and 1 (among negative real numbers) for 1?q<2, while the interval 0<q?1 corresponds to any real argument greater than 1 (positive real numbers). These two distinct intervals of the nonextensivity index q, also the expressions of the deformed functions associated with them, are related to one another through the relation (2−q), which is so far used to obtain the ordinary stationary distributions from the corresponding escort distributions, and vice versa in an almost ad hoc manner. This shows that the escort distributions are only a means of extending the interval of validity of the deformed functions to the one of ordinary, undeformed ones. Moreover, we show that, since the Tsallis entropy is written in terms of the q-logarithm and its argument, being the inverse of microstate probabilities, takes values equal to or greater than 1, the resulting stationary solution is uniquely described by the one obtained from the ordinary constraint. Finally, we observe that even the escort stationary distributions can be obtained through the use of the ordinary averaging procedure if the argument of the q-exponential lies in (−∞,0]. However, this case corresponds to, although related, a different entropy expression than the Tsallis entropy.     

Correlation and fluctuations in relativistic nucleus-nucleus collisions

Correlation and fluctuations are now well accepted analysis techniques in heavy-ion collisions at relativistic energies. At the current stage of RHIC exploration, matter in bulk and many of the physics questions about the final stage of collisions are addressed with the help of correlation techniques. In the present work after a general introduction to the underlying formalism to the exotic phenomena of correlation and fluctuations, discussion on various parameters disentangling dynamical fluctuations is presented. Analysis to investigate dynamical fluctuations and correlation is carried out in terms of F _q - and G _q -moments. A study of various other parameters involving multiplicity and pseudorapidity of relativistic charged particles produced in high energy nuclear interactions reveals the presence of correlation and fluctuations in particle production in these collisions. The experimental data on 14.5A GeV/c ²⁸Si-nucleus interactions has been analyzed. A parallel analysis of correlation free data generated using MC-RAND Monte Carlo code, UrQMD data and for the HIJING generated events has also been carried out.     

q-Gaussian approximants mimic non-extensive statistical-mechanical expectation for many-body probabilistic model with long-range correlations

We study a strictly scale-invariant probabilistic N-body model with symmetric, uniform, identically distributed random variables. Correlations are induced through a transformation of a multivariate Gaussian distribution with covariance matrix decaying out from the unit diagonal, as ρ/r ^α for r =1, 2, ..., N-1, where r indicates displacement from the diagonal and where 0 ⩽ ρ ⩽ 1 and α ⩾ 0. We show numerically that the sum of the N dependent random variables is well modeled by a compact support q-Gaussian distribution. In the particular case of α = 0 we obtain q = (1-5/3 ρ) / (1- ρ), a result validated analytically in a recent paper by Hilhorst and Schehr. Our present results with these q-Gaussian approximants precisely mimic the behavior expected in the frame of non-extensive statistical mechanics. The fact that the N → ∞ limiting distributions are not exactly, but only approximately, q-Gaussians suggests that the present system is not exactly, but only approximately, q-independent in the sense of the q-generalized central limit theorem of Umarov, Steinberg and Tsallis. Short range interaction (α > 1) and long range interactions (α < 1) are discussed. Fitted parameters are obtained via a Method of Moments approach. Simple mechanisms which lead to the production of q-Gaussians, such as mixing, are discussed.      

Circular-like maps: sensitivity to the initial conditions, multifractality and nonextensivity

Dissipative one-dimensional maps may exhibit special points (e.g., chaos threshold) at which the Lyapunov exponent vanishes. Consistently, the sensitivity to the initial conditions has a power-law time dependence, instead of the usual exponential one. The associated exponent can be identified with 1/(1-q), where q characterizes the nonextensivity of a generalized entropic form currently used to extend standard, Boltzmann-Gibbs statistical mechanics in order to cover a variety of anomalous situations. It has been recently proposed (Lyra and Tsallis, Phys. Rev. Lett. 80, 53 (1998)) for such maps the scaling law , where and are the extreme values appearing in the multifractal function. We generalize herein the usual circular map by considering inflexions of arbitrary power z, and verify that the scaling law holds for a large range of z. Since, for this family of maps, the Hausdorff dimension d_f equals unity for all z in contrast with q which does depend on z, it becomes clear that d_f plays no major role in the sensitivity to the initial conditions. Received 5 February 1999     

Production of charged pions, kaons and antikaons in relativistic C + C and C + Au collisions

Production cross-sections of charged pions, kaons and antikaons have been measured in C+C and C+Au collisions at beam energies of 1.0 and 1.8 AGeV for different polar emission angles. The kaon and antikaon energy spectra can be described by Boltzmann distributions whereas the pion spectra exhibit an additional enhancement at low energies. The pion multiplicity per participating nucleon M(π⁺)/<A _part> is a factor of about 3 smaller in C+Au than in C+C collisions at 1.0 AGeV whereas it differs only little for the C and the Au target at a beam energy of 1.8 AGeV. The K⁺ multiplicities per participating nucleon M(K⁺)/ <A _part> are independent of the target size at 1 AGeV and at 1.8 AGeV. The K^- multiplicity per participating nucleon M(K^-)/ <A _part> is reduced by a factor of about 2 in C+Au as compared to C+C collisions at 1.8 AGeV. This effect might be caused by the absorption of antikaons in the heavy target nucleus. Transport model calculations underestimate the K^-/K⁺ ratio for C+C collisions at 1.8 AGeV by a factor of about 4 if in-medium modifications of K-mesons are neglected. Received: 10 December 1999 / Accepted: 14 November 2000     

Energy fluctuation and correlation in Tsallis statistics

We investigate the fluctuation of the energy in the framework of Tsallis statistics and find the correlation plays an important role in energy fluctuations. In Tsallis statistics, the correlation is induced by the nonextensivity of Tsallis entropy and exists between particles even if the particles are dynamically independent. By taking the generalized ideal gas as an example, we get that when the particle number N is large enough, the relative fluctuation of the energy is proportional to 1/N instead of in Boltzmann statistics. Thus, the relative energy fluctuation is much smaller in Tsallis statistics than that in Boltzmann statistics. Besides, we demonstrate that the introduction of correlation between particle energies leads to smaller energy fluctuations in Tsallis statistics.     

Medium-modified average multiplicity and multiplicity fluctuations in jets

The energy evolution of average multiplicities and multiplicity fluctuations in jets produced in heavy-ion collisions is investigated from a toy QCD-inspired model. In this model, we use modified splitting functions accounting for medium-enhanced radiation of gluons by a fast parton which propagates through the quark–gluon plasma. The leading contribution of the standard production of soft hadrons is enhanced by a factor while next-to-leading order (NLO) corrections are suppressed by , where the parameter N _s >1 accounts for the induced soft gluons in the medium. Our results for such global observables are cross-checked and compared with their limits in the vacuum.     

Weakly nonextensive thermostatistics and the Ising model with long-range interactions

We introduce a new nonextensive entropic measure that grows like , where N is the size of the system under consideration. This kind of nonextensivity arises in a natural way in some N-body systems endowed with long-range interactions described by interparticle potentials. The power law (weakly nonextensive) behavior exhibited by is intermediate between (1) the linear (extensive) regime characterizing the standard Boltzmann-Gibbs entropy and (2) the exponential law (strongly nonextensive) behavior associated with the Tsallis generalized q-entropies. The functional is parametrized by the real number in such a way that the standard logarithmic entropy is recovered when . We study the mathematical properties of the new entropy, showing that the basic requirements for a well behaved entropy functional are verified, i.e., possesses the usual properties of positivity, equiprobability, concavity and irreversibility and verifies Khinchin axioms except the one related to additivity since is nonextensive. For , the entropy becomes superadditive in the thermodynamic limit. The present formalism is illustrated by a numerical study of the thermodynamic scaling laws of a ferromagnetic Ising model with long-range interactions. Received 24 May 2000     

Thermodynamic non-additivity in disordered systems with extended phase space

Non-additivity effects in coupled dynamic-stochastic systems are investigated. It is shown that there is a mapping of the replica approach to disordered systems with finite replica indexn on Tsallis non-extensive statistics, if the average thermodynamic entropy of the dynamic subsystem differs from the information entropy for the probability distribution in the stochastic subsystem. The entropic indexq is determined by the entropy difference ΔS. In the case of incomplete information, the entropic indexq=1−n is shown to be related to the degree of lost information.     

Nonextensive Statistical Mechanics Application to Vibrational Dynamics of Protein Folding

The vibrational dynamics of protein folding is analyzed in the framework of Tsallis statistics. We employ exact expressions for classical harmonic oscillators by considering the unnormalized constraints. As q→1, we show that these approximations agree with the result of Gaussian network model.     

Multiple ionization of argon in coincidence with projectile ions in 60–120 MeV Si q+-Ar collisions

A projectile ion-recoil ion coincidence technique has been employed to study the multiple ionization and the charge transfer processes in collisions of 60–120 MeV Si ^q+ (q = 4−14) ions with neutral argon atoms. The relative contribution of different ionization channels, namely; direct ionization, electron capture and electron loss leading to the production of slow moving multiply charged argon recoil ions have been investigated. The data reported on the present collision system result from a direct measurement in the considered impact energy for the first time. The total ionization cross-sections for the recoil ions are shown to scale as q ^1.7/E _p ^0.5 , where E _p is the energy in MeV of the projectile and q its charge state. The recoil fractions for the cases of total- and direct ionizations are found to decrease with increasing recoil charge state j. The total ionization fractions of the recoils are seen to depend on q and to show the presence of a ‘shell-effect’ of the target. Further, the fractions are found to vary as 1/j ² upto j = 8+. The average recoil charge state 〈j〉 increases slowly with q and with the number of lost or captured electrons from or into the projectile respectively. The projectile charge changing cross-sections σ _qq′ are found to decrease with increasing q for loss ionization and to increase with q for direct-and capture ionization processes respectively. The physics behind various scaling rules that are found to follow our data for different ionization processes is reviewed and discussed.