Advances in High Energy Physics

Volume 2018, Article ID 6384357, 15 pages

https://doi.org/10.1155/2018/6384357

## Multifractal Analysis of Charged Particle Multiplicity Distribution in the Framework of Renyi Entropy

^{1}Department of Physics, New Alipore College, L Block, New Alipore, Kolkata 700053, India^{2}Institute of Space Science, Bucharest, Romania

Correspondence should be addressed to Swarnapratim Bhattacharyya; moc.oohay@mitarp_anraws

Received 3 March 2018; Accepted 22 May 2018; Published 26 June 2018

Academic Editor: Sakina Fakhraddin

Copyright © 2018 Swarnapratim Bhattacharyya et al. 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

A study of multifractality and multifractal specific heat has been carried out for the produced shower particles in nuclear emulsion detector for ^{16}O-AgBr, ^{28}Si-AgBr, and ^{32}S-AgBr interactions at 4.5AGeV/c in the framework of Renyi entropy. Experimental results have been compared with the prediction of Ultra-Relativistic Quantum Molecular Dynamics (UrQMD) model. Our analysis reveals the presence of multifractality in the multiparticle production process in high energy nucleus-nucleus interactions. Degree of multifractality is found to be higher for the experimental data and it increases with the increase of projectile mass. The investigation of quark-hadron phase transition in the multiparticle production in ^{16}O-AgBr, ^{28}Si-AgBr, and ^{32}S-AgBr interactions at 4.5 AGeV/c in the framework of Ginzburg-Landau theory from the concept of multifractality has also been presented. Evidence of constant multifractal specific heat has been obtained for both experimental and UrQMD simulated data.

#### 1. Introduction

The study of nonstatistical fluctuations and correlations in relativistic and ultra-relativistic nucleus-nucleus collisions has become a subject of major interest among the particle physicists. Bialas and Peschanski [1, 2] proposed a new phenomenon called intermittency to study the nonstatistical fluctuations in terms of the scaled factorial moment. In high energy physics intermittency is defined as the power law behavior of scaled factorial moment with the size of the considered phase space [1, 2]. This method has its own advantage that it can extract nonstatistical fluctuations after extricating the normal statistical noise [1, 2]. The study of nonstatistical fluctuations by the method of scaled factorial moment leads to the presence of self-similar fractal structure in the multiparticle production of high energy nucleus-nucleus collisions [3]. The self-similarity observed in the power law dependence of scaled factorial moments reveals a connection between intermittency and fractality. The observed fractal structure is a consequence of self-similar cascade mechanism in multiparticle production process. The close connection between intermittency and fractality prompted the scientists to study fractal nature of multiparticle production in high energy nucleus-nucleus interactions. To get both qualitative and quantitative idea concerning the multiparticle production mechanism fractality study in heavy-ion collisions is expected to be very resourceful.

#### 2. Fractals-Multifractals and Monofractals

The term “fractal” was coined by Mandelbrot [4] from the Latin word fractus which means broken or fractured. Mandelbrot introduced the new geometry called the fractal geometry to look into the world of apparent irregularities. A fractal pattern is one that scales infinitely to reproduce itself such that the traditional geometry does not define it. In other words a fractal is generally a rough or fragmented geometrical shape that can be split into parts, each of which is at least approximately reduced size copy of the whole [4]. Sierpinski triangle, Koch snow flack, Peano curve, Mandelbrot set, and Lorentz attractor are the well-known mathematical structures that exhibit fractal geometry [4]. Fractals also describe many real world objects such as clouds, mountains, coastlines, etc.; those do not corresponds to simple geometrical shape [4]. Fractals can be classified into two categories: multifractals and monofractals [4]. Multifractals are complicated self-similar objects that consist of differently weighted fractals with different noninteger dimensions. The fundamental characteristic of multifractality is that the scaling properties may be different in different regions of the systems [4]. Monofractals are those whose scaling properties are the same in different regions of the system [4]. As the scaling properties are dissimilar in different parts of the system, multifractal systems require at least more than one scaling exponent to describe the scaling behavior of the system [5]. A distinguishing feature of the processes that have multifractal characteristics is that various associated probability distributions display power law properties [6, 7]. Multifractal theory is essentially rooted in probability theory though draws on complex ideas from each of physics, mathematics, probability theory, and statistics [6]. Apart from multiparticle production in high energy physics, multifractal analysis has proved to be a valuable method of capturing the underlying scaling structure present in many types of systems including diffusion limited aggregation [8–10], fluid flow through random porous media [11], atomic spectra of rare-earth elements [12], cluster-cluster aggregation [13], and turbulent flow [14]. In physiology, multifractal structures have been found in heart rate variability [15] and brain dynamics [16]. Multifractal analysis has been helpful in distinguishing between healthy and pathological patients [17]. Multifractal measures have also been found in man-made phenomena such as the Internet [18], art [19], and the stock market [20–22]. Multifractals have also been used in a wide range of application areas like the description of dynamical systems, rainfall modelling, spatial distribution of earthquakes and insect populations, financial time series modelling, and Internet traffic modelling [6, 7].

It should be mentioned here that the most important property of fractals is their dimensions [4, 6, 7]. Fractal dimension is used to describe the size of the fractal sets [4, 6, 7]. For example, the dimension of an irregular coastline may be greater than one but less than two, indicating that it is not like a simple line and has space filling characteristics in the plane. Likewise, the surface area of a snowflake may be greater than two but less than three, indicating that its surface is more complex than regular geometrical shapes and is partially volume filling [4, 6, 7]. Fractal dimension can be calculated by taking the limit of the quotient of the log change in object size and the log change in measurement scale, as the measurement scale approaches to zero [4]. The differences came in what is meant by the measurement scale and how to get an average number out of many different parts of the geometrical objects [4, 6, 7]. Fractal dimension quantifies the static geometry of an object [4].

Generalized fractal dimension is a well-known parameter which reflects the nature of fractal structure [4]. From the dependence of generalized fractal dimension on the order q a distinguishable characterization of fractality is possible [4]. Decrease of generalized fractal dimension with the order of moment signals the presence of multifractality. On the other hand if remains constant with the increase of order q monofractality occurs. It has been pointed out in [23] that if Quark-Gluon-Plasma (QGP) state is created in hadronic collisions, a phase transition to hadronic matter will take place. The hadronic system will show monofractality in contrast to an order dependence of generalized fractal dimension , when a cascade process occurs [23].

Hwa [24] was the first to provide the idea of using multifractal moments , to study the multifractality and self-similarity in multiparticle production. According to the method enunciated by Hwa [24] if the particle production process exhibits self-similar behavior, the moments show a remarkable power law dependence on phase space bin size. However, if the multiplicity is low, the moments are dominated by statistical fluctuations. In order to suppress the statistical contribution, a modified form of moments in terms of the step function was suggested by Hwa and Pan [25]. Takagi [26] also proposed a new method called Takagi moment method ( moment) for studying the fractal structure of multiparticle production. Both the and the techniques have been applied extensively to analyze several high energy nucleus-nucleus collision data [27–35]. Very recently some sophisticated methods have also been applied to study the fractal nature of multiparticle production process [36–45].

#### 3. Entropy and Fractality

In high energy nucleus-nucleus collisions, entropy measurement of produced shower particles may provide important information in studying the multiparticle production mechanism [46]. In high energy collisions, entropy is an important parameter and it is regarded as the most significant characteristic of a system having many degrees of freedom [46]. As entropy reflects how the effective degrees of freedom change from hadronic matter at low temperature to the quark-gluon plasma state at high temperature, it is regarded as a useful probe to study the nature of deconfinement phase transition [46]. Entropy plays a key role in the evolution of the high temperature quark-gluon plasma in ultra-relativistic nucleus-nucleus interactions in Relativistic Heavy-Ion Collider (RHIC) experiments and in Large Hadron Collider (LHC) experiments [46]. In nucleus-nucleus collisions, entropy measurement can be used to not only search for the formation of Quark-Gluon-Plasma (QGP) state but it may also serve as an additional tool to investigate the correlations and event-by-event fluctuations [47]. Different workers have investigated the evolution of entropy in high energy nucleus-nucleus collisions at different times [48–64]. The entropy of produced particles can easily be calculated from the multiplicity distribution of the data. If is the probability of producing n particles in a high energy interaction the entropy S is defined by the relation [65]. Mathematically this entropy is called the Shannon entropy [66].

Apart from studying the well-known Shannon entropy, people are interested to explore the hidden physics of Renyi entropy [66–69] as well, mainly motivated by the inspiration of A. Bialas and W. Czyż [51, 64, 70]. According to C.W Ma and Y.G. Ma [71] the difference between qth order Renyi entropy and order Renyi entropy is found just to be a q dependent constant but which is very sensitive to the form of probability distribution. Renyi entropy can play a potential role to investigate the fractal characteristics of multiparticle production process [72, 73]. The advantage of this method to study the fractal properties of multiparticle production process is that it is not related to the width and resolution of the phase space interval [72, 73]. This method can be applied to events having higher as well as lower multiplicity. This method never suffers from the drawback of lower statistics.

In terms of the probability of multiplicity distribution , the q th order Renyi entropy can be defined as [51, 64, 70]If = , then (1) can be written asGeneralized fractal dimension can be evaluated from the concept of Renyi entropy according to the relationwhere is the central rapidity value in the centre of mass frame and is given by Here is the centre of mass energy of the concerned collision process, is the rest mass of pions, and denotes the average number of participating nucleons.

The generalized fractal dimension is related to the anomalous fractal dimension by a simple mathematical relation [74]Goal of our present study is to carry out an investigation of multifractality and multifractal specific heat in shower particle multiplicity distribution from the concept of Renyi entropy measurements in ^{16}O-AgBr, ^{28}Si-AgBr, and ^{32}S-AgBr interactions at 4.5AGeV/c. We have compared our experimental results with the prediction of Ultra-Relativistic Quantum Molecular Dynamics (UrQMD) model. Importance of this study is that so far very few attempts have been made to explore the presence of multifractality in multiparticle production process in the framework of higher-order Renyi entropy in high energy nucleus-nucleus interactions.

#### 4. Experimental Details

In order to collect the data used for the present analysis, stacks of NIKFI-BR2 emulsion pellicles of dimensions 20cm10 cm600 were irradiated by the ^{16}O, ^{28}Si, and ^{32}S beam at 4.5 AGeV/c obtained from the Synchrophasotron at the Joint Institute of Nuclear Research (JINR), Dubna, Russia [75–78]. According to Powell [79], in nuclear emulsion detector particles emitted and produced from an interaction are classified into four categories, namely, the shower particles, the grey particles, the black particles, and the projectile fragments. Details of scanning and measurement procedure of our study along with the characteristics of these emitted and produced particles in nuclear emulsion can be found from our earlier publications [75–78].

One of the problem encountered in interpreting results from nuclear emulsion is the nonhomogeneous composition of emulsion which contain both light (H,C,N, and O) and heavy target nuclei (Ag, Br). In emulsion experiments it is very difficult to identify the exact target nucleus [75]. Based on the number of heavy tracks () total number of inelastic interactions can be divided into three broad target groups H, CNO, and AgBr in nuclear emulsion [75]. Detailed method of target identification has been described in our earlier publication [75]. For the present analysis we have not considered the events which are found to occur due to collisions of the projectile beam with H and CNO target present in nuclear emulsion. Our analysis has been carried out for the interactions of three different projectile ^{16}O, ^{28}Si, and ^{32}S at 4.5 AGeV/c with the AgBr target only. Applying the criteria of selecting AgBr events ( >8) we have chosen 1057 events of ^{16}O-AgBr interactions, 514 events of ^{28}Si-AgBr, and 434 events of ^{32}S-AgBr interactions at 4.5 AGeV/c [75]. Our analysis has been performed on the shower tracks only. We have calculated the average multiplicity of shower tracks in each interaction and presented the values in Table 1 [75–78].