Advances in High Energy Physics

Volume 2018, Article ID 2136908, 11 pages

https://doi.org/10.1155/2018/2136908

## Entropy and Multifractality in Relativistic Ion-Ion Collisions

Department of Physics, Aligarh Muslim University, Aligarh 202002, India

Correspondence should be addressed to Shakeel Ahmad; hc.nrec@damha.leekahs

Received 31 August 2017; Revised 5 January 2018; Accepted 11 March 2018; Published 11 April 2018

Academic Editor: Raghunath Sahoo

Copyright © 2018 Shaista Khan and Shakeel Ahmad. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. The publication of this article was funded by SCOAP^{3}.

#### Abstract

Entropy production in multiparticle systems is investigated by analyzing the experimental data on ion-ion collisions at AGS and SPS energies and comparing the findings with those reported earlier for hadron-hadron, hadron-nucleus, and nucleus-nucleus collisions. It is observed that the entropy produced in limited and full phase space, when normalized to maximum rapidity, exhibits a kind of scaling which is nicely supported by Monte Carlo model HIJING. Using Rényi’s order information entropy, multifractal characteristics of particle production are examined in terms of generalized dimensions, . Nearly the same values of multifractal specific heat, , observed in hadronic and ion-ion collisions over a wide range of incident energies suggest that the quantity might be used as a universal characteristic of multiparticle production in hadron-hadron, hadron-nucleus, and nucleus-nucleus collisions. The analysis is extended to the study of spectrum of scaling indices. The findings reveal that Rényi’s order information entropy could be another way to investigate the fluctuations in multiplicity distributions in terms of spectral function , which has been argued to be a convenient function for comparison sake not only among different experiments but also between the data and theoretical models.

#### 1. Introduction

The main objective of investigating the collisions of heavy nuclei at relativistic energies is to understand the nature of phase transition from normal hadronic matter to quark-gluon plasma (QGP) [1]. Dedicated experiments have been carried out at AGS and RHIC at BNL and SPS and LHC at CERN to search for the QGP phase and explore the QCD phase diagram. Heavy-ion collision experiments provide a unique opportunity to test the predictions of QCD and at the same time to understand the soft processes involved, which dominate even at LHC energies, as well as the hard scattering or the small cross-sectional physics [2–4]. Relativistic nucleus-nucleus (AA) collisions serve as a mean to study the novel regime of QCD where parton densities are high enough while the strong coupling constant between the partons is small and decreases further with decreasing interpartonic distances [2]. The parton densities in the early stage of the collision can be related to the density of the produced charged hadrons in the final state. Dominance of hard process (jets and minijets) in the hadron production increases with increasing collision energy and hence provides a unique opportunity to study the interplay between effects. In this scenario, the perturbative QCD (pQCD) lends a good basis for high energy dynamics and has achieved significant success in describing the hard processes involved in high energy collisions [2, 5]. Hadronic-jet production in annihilations [6, 7] and large jet production in hadron-hadron (hh) collisions [8, 9] are the examples of these hard processes. However, in soft processes like hadron production with sufficiently low in hadronic and heavy-ion collisions, the interactions become so strong that the pQCD is not applicable anymore [2]. Due to incapability of pQCD in this regime, phenomenological models based on experimental inputs have been proven to be an alternative tool to understand the dynamics of particle production in AA collisions.

Relativistic charged particle multiplicity is the simplest observable and by studying its distribution for a given data sample, information on soft QCD processes as well as on hard scattering can be extracted [3]. Furthermore, the charged hadron multiplicity of an event is taken as a direct measure of its inelasticity and can describe important features of particle production [3]. Investigations involving multiparticle production in hh, hA, and AA collisions at relativistic energies have been carried out by numerous groups during the last four decades [10–15]. However, a complete understanding of particle production mechanism still remains elusive. Multiplicity distributions (MD) of relativistic charged particles produced in hh collisions have been observed to deviate from a Poisson distribution and are expected to provide information regarding the underlying production mechanism [13, 16]. Asymptotic scaling of MD in hh collisions referred to as the KNO scaling [17], predicted in 1972, was regarded as a useful phenomenological framework for comparing the MD at different energies ranging from GeV to ISR energies [18]. It was, however, pointed out [19, 20] that KNO scaling is not strictly followed for inelastic hh collisions. This scaling law was observed to breakdown when collision energy reached SPS range [13, 16, 18, 21]. After the observations of KNO scaling violation in collisions at = 540 GeV, it was remarked that the observed scaling up to ISR energies was approximate and accidental [21]. A new empirical regularity, in place of KNO scaling, was then proposed [21] to predict the multiplicity distributions at different energies. It was shown that MD at different energies in full and limited rapidity () windows may be nicely reproduced by negative binomial distributions (NBD) [18, 21, 22]. These observations lead to revival of interest in investigations involving MD and new scaling laws. Simak et al. [23], by introducing a new variable, the information entropy, showed that MD of charged particles produced in full and limited phase space in hh collisions exhibits a new type of scaling law in the energy range, to 900 GeV.

Sinyukov and Akkelin [24] have proposed a method to estimate entropy of thermal pions in AA collisions and have studied the average phase space densities and entropy of such pions against their multiplicities and beam energies. Their findings apparently suggest the presence of deconfinement and chiral phase transition in relativistic AA collisions. Moreover at RHIC energies, entropy per unit rapidity at freeze-out has been extracted with minimal model dependence from the available measurements of particle yields, spectra, and source sizes, determined from two-particle interferometry [25]. The extracted entropy per unit rapidity was observed to be consistent with the lattice gauge theory for thermalized QGP with an energy density calculated from the transverse energy production at RHIC energies.

Analyses of the experimental data on , , and collisions over a wide energy range (up to = 900 GeV) carried out by several workers [23, 26, 27] indicate that entropy increases with beam energy while the entropy per unit rapidity appears to be an energy independent quantity. These results indicate the presence of entropy scaling up to a few TeV energy. Presence of a similar scaling behaviour in collisions at LHC energies has also been reported by Mizoguchi and Biyajima [28] and Das et al. [11, 13]. Analyses of AA collision data at AGS and SPS energies carried out by other workers [10, 15, 29, 30] too suggest that entropy produced in limited pseudorapidity () windows when normalized to the maximum rapidity is, essentially, independent of projectile and target mass as well as the beam energy, indicating the presence of entropy scaling.

The occurrence of unusual large particle density fluctuations in narrow phase space bins, observed in cosmic ray JACEE events [31] and in accelerator experiments [32, 33], have generated considerable interest in the study of nonlinear phenomena in hadronic and heavy-ion collisions. Such fluctuations may be taken as an indication of a phase transition from ordinary hadronic matter to QGP, predicted by QCD to occur in relativistic AA collisions. These fluctuations are also envisaged to arise either due to minijets produced at very high energies or (and) because of some other collective phenomena [34]. Such rare fluctuations of dynamical origin are to be identified and extracted from the statistical ones. Various methods for the identification of these fluctuations have been proposed which estimate the total deviation of the measured rapidity distribution from an idealized smooth distribution [34]. The analysis is based on the estimation of corresponding statistical probability through Monte Carlo (MC) simulations. The method of scaled factorial moments (SFMs), proposed by Bialas and Peschanski [35], has been proven to be well suited for not only to search for the dynamical fluctuations in narrow phase space bins but also to investigate the pattern of fluctuations which could lead to a physical interpretation of their origin [35]. The concept of SFMs was put forward in analogy with the phenomenon referred to as the “intermittency” in hydrodynamics of turbulent fluid flow [34]. This phenomenon is characterized by the presence of fluctuations in a small part of the available phase space. The method of SFMs was first applied to the pseudorapidity distribution of a single high multiplicity JACEE event [31] and the findings indicated the presence of intermittent pattern (large particle density fluctuations in narrow pseudorapidity bins) [35]. SFMs analysis, since its introduction, has been widely used to search for the nonlinear phenomena in hadronic and ion-ion collisions. It has been observed [36–39] that the presence of such a nonlinear phenomena might be rare but not impossible [40]. Such fluctuations, if not arising due to the statistical reasons, are envisaged to occur due to the dynamical correlations among the produced particles. These correlations might arise due to the phase transition from QGP to normal hadronic matter. As pointed out by Bialas and Peschanski [35], if the intermittency exists, the SFMs of multiplicity distributions should exhibit a power law dependence of the rapidity-bin width, as . Such a study would help understand the chaotic behaviour of rapidity distribution on event-by-event (ebe) basis instead of investigating the average phenomena [41].

The observations of intermittency in annihilation [42], hh [43], hA [34], and AA collisions [41] attracted a great deal of attention towards the investigations involving the power law behaviour of SFMs of the form where the exponent increases with order of the moment [41].

Although the intermittency analysis in terms of SFMs has been successfully applied to the hadronic and heavy-ion collisions, yet the dynamical explanation of its origin in some cases is not clear [44]. The concept of self-similarity is closely related to the fractal theory [45, 46] which is a natural consequence of the cascading mechanism prevailing in the multiparticle production. A formalism for treating the fractal dimensions and its generalization has been developed and applied to the study of turbulent fluids and other transitions to chaos [41, 47]. However, in the case of multiparticle production, the dynamics is not well known and one has to discover the proper dynamics that gives rise to the chaotic structure. The fractal dimensions, in multiparticle production, were first studied by Carruthers and Duong-Van [48]. Later on Dermin [49] suggested the study of correlation dimensions, while Lipa and Buschbeck [50] considered other generalized dimensions. However, in none of these investigations a formalism for systematic studies of fractal properties has been presented. Considering inelastic collisions as purely geometrical objects with noninteger dimensions, a formalism for treating the fractal dimensions and its generalization was developed and successfully applied to the study of intermittent behaviour and other transitions to chaos [44, 47, 51–53]. In order to reveal the multifractal nature of multiparticle production, Hwa [41], Chiu and Hwa [54], and Florkowski and Hwa [55] introduced moments, which can be estimated from the particle multiplicities in limited rapidity windows that might lead to inferences on the nature of the dimensions, , that are the generalizations of the fractal dimensions for multifractal sets. Using the moments properties of scaling indices spectrum may be investigated which, in turn, would indicate as to how it can provide effective means to describe a highly nonuniform rapidity distribution.

As the studies involving AA collisions at relativistic energies are concerned, the main goal is to study the properties of strongly interacting matter under extreme conditions of nuclear density and temperature, where formation of quark-gluon plasma (QGP) is predicted to take place [10, 15, 56, 57]. Fluctuations in physical observables in AA collisions are regarded as one of the important signals, for QGP formation because of the idea that, in many body systems, phase transition may result in significant changes in the quantum fluctuations of an observable from its average behaviour [15, 29, 56]. For example, when a system undergoes a phase-transition, heat capacity changes abruptly, whereas the energy density remains the smooth function of temperature [10, 15, 25, 58, 59]. Entropy is regarded yet another important characteristics of a system with many degrees of freedom [15, 60–63]. Processes in which particles are produced may be considered as the so-called* dynamical systems* [60–64] in which entropy is generally produced. Systematic measurements of local entropy produced in AA collisions may provide direct information about the internal degrees of freedom of the QGP medium and its evolution [10, 15, 27]. It has been pointed out [65] that in high energy collisions particle production occurs on the maximum stochasticity; that is, they should follow the maximum entropy principle. This type of stochasticity may also be quantified in terms of information entropy which may be regarded as a natural and more general parameter to measure the chaoticity in branching processes [66].

Bialas et al. [60, 61, 67] proposed that Rényi entropies may also be used as a tool for studying the dynamical systems and are closely related to the thermodynamic entropy of the system, the Shannon entropy. Bialas et al. [68] have also suggested that Rényi’s entropies may serve as a useful tool for examining the correlation among the particles produced in high energy collisions. Furthermore, the generalization of Rényi’s order- information entropy contains information on the multiplicity moments that can be used to investigate the multifractal characteristics of particle production [14, 69, 70]. It should be mentioned that this method of multifractal studies is not related to the phase space bin-width or the detector resolution but to the collision energy [69]. It is, therefore, considered worthwhile to carry out a well focused study of entropy production and subsequent scaling in AA collisions by analyzing the several sets of experimental data on AA collisions at AGS and SPS energies. Rényi’s information entropies are also estimated to investigate the multifractal characteristics of multiparticle production. Moreover, the analysis is further extended to study the fractal dimensions, multifractal spectrum, and their dependence on the energy and mass of the projectile.

#### 2. Details of Data

Six sets of events produced in collisions of ^{16}O, ^{28}Si, ^{32}S, and ^{197}Au beams with AgBr group of nuclei in emulsion at AGS and SPS energies, available in the laboratory, are used in the present study. Details of these samples are presented in Table 1. These events are taken from the emulsion experiments performed by EMU01 Collaboration [71–74]. The other relevant details of the data, like selection criteria of events, classification of tracks, extraction of AgBr group of interactions, method of measuring the emission angle, of relativistic charged particles, and so on, may be found elsewhere [10, 15, 71, 75–77].