Multifractal analysis of charged particle multiplicity distribution in the framework of Renyi entropy

A study of multifractality and multifractal specific heat has been carried out for the produced shower particles in nuclear emulsion detector for 16O-AgBr, 28Si-AgBr and 32S-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 16O-AgBr, 28Si-AgBr and 32S-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.


Introduction
The study of non statistical fluctuations and correlations in relativistic and ultra-relativistic nucleus-nucleus collisions have become a subject of major interest among the particle physicists. Bialas and Peschanski [1][2] proposed a new phenomenon called intermittency to study the non-statistical 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 non-statistical fluctuations after extricating the normal statistical noise [1][2]. The study of non-statistical fluctuations by the method of scaled factorial moment leads to the presence of self-similar fractal structure in the multi-particle 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.
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][9][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][21][22].
Multifractals has 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 modeling [6][7].
It should be mentioned here that the most important property of fractals are 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 4 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 G q , 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 G q moments show a remarkable power law dependence on phase space bin size. However, if the multiplicity is low, the G q moments are dominated by statistical fluctuations. In order to suppress the statistical contribution, a modified form of G q 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 (T q moment) for studying the fractal structure of multiparticle production. Both the G q and the T q techniques have been applied extensively to analyze several high-energy nucleus-nucleus collision data [27][28][29][30][31][32][33][34][35]. Very recently some sophisticated methods have also been applied to study the fractal nature of multiparticle production process [36][37][38][39][40][41][42][43][44][45][46][47].

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 [48].
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 [48]. As entropy reflects how the effective degrees of freedom changes 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 [48]. 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 [48]. In nucleus-nucleus collisions, entropy measurement can be used not only to 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 [49].
Mathematically this entropy S is called the Shannon entropy [68].
Apart from studying the well known Shannon entropy, people are interested to explore the hidden physics of Renyi entropy [68][69][70][71] as well, mainly motivated by the inspiration of A.
Białas and W. Czyz˙ [66,[72][73]. According to C.W Ma and Y.G. Ma [74] the difference between qth order Renyi entropy and 1 st 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 [75][76]. 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 [75][76]. 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  Where 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, denotes the average number of participating nucleons.
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 6 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 nucleusnucleus interactions.

Experimental Details
In order to collect the data used for the present analysis, stacks of NIKFI-BR2 emulsion pellicles of dimension 20cm 10 cm 600 were irradiated by the 16 O, 28 Si and 32 S beam at 4.5 AGeV/c obtained from the Synchrophasatron at the Joint Institute of Nuclear Research (JINR), Dubna, Russia [78][79][80][81]. According to the Powell [82], 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 [78][79][80][81].
One of the problem encountered in interpreting results from nuclear emulsion is the nonhomogeneous composition of emulsion which contain both light (H,C,N,O) and heavy target nuclei (Ag, Br). In emulsion experiments it is very difficult to identify the exact target nucleus [78]. Based on the number of heavy tracks (N h ) total number of inelastic interactions can be divided into three broad target groups H, CNO and AgBr in nuclear emulsion [78]. Detailed method of target identification has been described in our earlier publication [78]. 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

Analysis and Results
In a very recent paper [83] we have investigated the Renyi entropy of second order of shower particles using 16 O, 28 Si and 32 S projectiles on interaction with AgBr and CNO target present in nuclear emulsion at an incident momentum of 4.5 AGeV/c. In this paper we have extended our analysis of Renyi entropy to the study of fractality in multiparticle production of 16 O-AgBr, 28 Si-AgBr and 32 S-AgBr interactions at 4.5AGeV/c.
In order to study the fractal nature of multiparticle production from the concept of Renyi entropy we have calculated the Renyi entropy values of order q=2-5 from the relation (1) and (2) 16 O-AgBr, 28 Si-AgBr and 32 S-AgBr interactions. Error bars drawn to every experimental point are statistical errors only. Using equation (3) and (4)  suggesting the presence of multifractality in multipion production mechanism. Presence of multifractality during the production of shower particles indicates the occurrence of cascade mechanism in particle production process. From the A non zero value of the implies the multifractal behaviour [84]. We have applied this theory to our study in order to quantify the fractal nature of shower particle production. We have plotted the variation of against the order number in figure 3 for 16 O-AgBr, 28 Si-AgBr and 32 S-AgBr interactions. The calculated values of the slope parameter have been presented in table 6 for our data. The value of signifies the degree of multifractality. From the table it may be seen that for all the interactions the value is greater than zero. This reconfirms the multifractal nature of multiparticle production mechanism. Moreover, value characterizing the degree of multifractality is found to depend on the mass number of the projectile beam. Degree of multifractality is found to increase with the increase of projectile mass as evident from table 6.
R.C Hwa suggested that [85] from the concept of multifractality a qualitative and quantitative investigation of quark-hadron phase transition in high energy nucleus-nucleus collisions is possible. In analogy with the photo count problem at the onset of lasing in non linear optics, the coherent state description in high energy nucleus-nucleus interactions can be used in the frame work of Ginzburg-Landau theory [85][86]. A quantity in terms of the ratio of higher order anomalous fractal dimension to the second order anomalous fractal dimension can be defined by the following relation [85][86] = − ……………………… According to Ginzburg-Landau model [86] is related with (q-1) by the relation = − …………………….…. . The relation and relation are found to be valid for all systems which can be described by the Ginzburg-Landau (GL) theory and also is independent of the underlying dimension of the parameters of the model [86]. If the value of the scaling exponent is equal to or close enough (within the experimental error) to 1.304 then a quark-hadron phase transition is expected for the experimental data [86]. If the measured value of is different from the critical value 1.304 considering the experimental errors then the possibility of the quark-hadron phase transition has to be ruled out [86].   16 O-AgBr and 28 Si-AgBr interactions are found to be lower than the critical value 1.304 while for 32 S-AgBr interaction the critical exponent is higher than the critical value signifying the absence of quark-hadron phase transition in our data.
Interpretation of multifractality from the thermodynamical point of view allows us to study the fractal properties of stochastic processes with the help of standard concept of thermodynamics. In thermodynamics the constant specific heat approximation is widely applicable in many important cases; for example, the specific heat of gases and solids is constant, independent of temperature over a larger or smaller temperature interval [87]. This approximation is also applicable to multifractal data of multiparticle production processes proposed by Bershadskii [88]. Bershadskii pointed out that [89] regions of the temperature where the constant-specific-heat approximation is applicable are usually far away from the phase-transition regimes. For the considered interactions the phase transition like phenomena can occur in the vicinity of q=0, where q is the inverse of temperature. Such situation is expected at the onset of chaos of the dynamical attractors [89]. Barshadskii argued that [88] multiparticle production in high energy nucleus-nucleus collisions is related to phase-transition-like phenomena [3]. He introduced a multifractal Bernoulli distribution which appears in a natural way at the morphological phase transition from monofractality to multifractality. Multifractal Bernoulii distribution plays an important role in multiparticle production at higher energies. It has been pointed out that [90] multifractal specific heat can be derived from the relation = ∞ + − if the monofractal to multifractal transition is governed by the Bernoulli distribution. The slope of the linear best fit curve showing the variation of against − has been designated as multifractal specific heat. The gap at the multifractal specific heat at the multifractality to monofractality transition allows us to consider this transition as a thermodynamic phase transition [91][92].
We have studied the variation of against − for 16 O-AgBr, 28 Si-AgBr and 32 [93,94]. In our earlier papers [81,83] we have utilized the UrQMD model to simulate 16 O-AgBr, 28 Si-AgBr and 32 S-AgBr interactions at 4.5 AGeV/c. As described in our previous papers [81,83] we have generated a large sample of events using the UrQMD code (UrQMD 3.3p1) for 16 O-AgBr, 28 Si-AgBr and 32 S-AgBr interactions at 4.5AGeV/c. We have also calculated the average multiplicities of the shower tracks for all the three interactions in case of the UrQMD data sample [81]. This observation signifies the presence of stronger multifractality for the experimental data.
We have also studied the Ginzburg-Landau phase transition with UrQMD simulated events of 16 O-AgBr, 28 Si-AgBr and 32 S-AgBr interactions. The variation of with (q-1) for the UrQMD simulated data have been presented in figure 4,in figure 5 and in figure 6 in case of 16 O-AgBr, 28 Si-AgBr and 32  We have studied the variation of against − in figure 7 for the UrQMD data sample of 16 O-AgBr, 28 Si-AgBr and 32 S-AgBr interactions in order to calculate the multifractal specific heat. From the slope of the best linear behavior of the plotted points the multifractal specific heat of the shower particles for the UrQMD data sample have been evaluated and presented in table 8 for all the three interactions. From the table it may be seen that the values of the multifractal specific heat for the experimental and simulated data agrees reasonably well with each other. This observation signifies the constancy of multifractal specific heat for the UrQMD simulated events also.

Conclusions and Outlook
In this paper we have presented an analysis of multifractality and multifractal specific heat in the frame work of Renyi entropy analysis for the produced shower particles in nuclear emulsion detector for 16 O-AgBr, 28 Si-AgBr and 32  It is true that there are many papers available in the literature where presence of multifractality has been tested experimentally in multiparticle production in high energy nucleus-nucleus interactions by different methods. But the method adopted in this paper to study multifractality seems to be simple and interesting and in this regard our study deserves attention. The observed multifractal behavior of the produced shower particles may be viewed as an experimental fact.     28 Si-AgBr and 32 S-AgBr interactions at 4.5AGeV/c.  Table 4 represents the values of Shannon Entropy S and Information dimension for all the three interactions in case of the experimental as well as the UrQMD simulated data.  28 Si-AgBr and 32 S-AgBr interactions at 4.5AGeV/c.  Table 6 represents the value characterizing the degree of multifractality obtained from the plot with order number q for 16 O-AgBr, 28 Si-AgBr and 32 S-AgBr interactions at 4.5AGeV/c in case of experimental and UrQMD simulated events.  Table 7 represents the values of the critical value for the Ginzburg-Landau phase transition for 16 O-AgBr, 28 Si-AgBr and 32 S-AgBr interactions at 4.5 AGeV/c for experimental and UrQMD simulated events calculated from the concept of Renyi entropy.  Table 8 represents the values of multifractal specific heat of the produced shower particles in 16 O-AgBr, 28 Si-AgBr and 32 S-AgBr interactions at 4.5AGeV/c in case of experimental and UrQMD simulated events.  S-AgBr---UrQMD Figure 2 represents the variation of generalized fractal dimension with order number q for 16 O-AgBr, 28 Si-AgBr and 32 S-AgBr interactions at 4.5AGeV/c in case of experimental and UrQMD simulated events.  Figure 3 represents the variation of with order number q for 16 O-AgBr, 28 Si-AgBr and 32 S-AgBr interactions at 4.5AGeV/c in case of experimental and UrQMD simulated events.