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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.

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 [

The term “fractal” was coined by Mandelbrot [

It should be mentioned here that the most important property of fractals is their dimensions [

Generalized fractal dimension

Hwa [

In high energy nucleus-nucleus collisions, entropy measurement of produced shower particles may provide important information in studying the multiparticle production mechanism [

Apart from studying the well-known Shannon entropy, people are interested to explore the hidden physics of Renyi entropy [

In terms of the probability of multiplicity distribution

The generalized fractal dimension ^{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.

In order to collect the data used for the present analysis, stacks of NIKFI-BR2 emulsion pellicles of dimensions 20cm^{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 [

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 [^{16}O, ^{28}Si, and ^{32}S at 4.5 AGeV/c with the AgBr target only. Applying the criteria of selecting AgBr events (^{16}O-AgBr interactions, 514 events of ^{28}Si-AgBr, and 434 events of ^{32}S-AgBr interactions at 4.5 AGeV/c [

This represents the average multiplicities of the shower particles for all the interactions in case of the experimental and the UrQMD data.

| | |
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| | |

^{16}O-AgBr | 18.05 | 17.79 |

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^{28}Si-AgBr | 23.62 | 27.55 |

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^{32}S-AgBr | 28.04 | 30.84 |

In a very recent paper [^{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 relations (^{16}O-AgBr, ^{28}Si-AgBr, and ^{32}S-AgBr interactions at 4.5AGeV/c have been presented in Table ^{16}O-AgBr, ^{28}Si-AgBr, and ^{32}S-AgBr interactions. Error bars drawn to every experimental point are statistical errors only. Using (^{16}O-AgBr, ^{28}Si-AgBr, and ^{32}S-AgBr interactions. Calculated values of generalized fractal dimension ^{16}O-AgBr, ^{28}Si-AgBr, and ^{32}S-AgBr interactions have been presented in Table ^{16}O-AgBr and ^{28}Si-AgBr interactions. But for ^{32}S-AgBr interactions the ^{16}O-AgBr, ^{28}Si-AgBr, and ^{32}S-AgBr interactions.

This represents the experimental and UrQMD simulated values of Renyi entropy ^{16}O-AgBr, ^{28}Si-AgBr, and ^{32}S-AgBr interactions at 4.5AGeV/c.

Interactions | Order | | |
---|---|---|---|

| 2 | 3.47 | 3.05 |

3 | 3.41 | 2.99 | |

4 | 3.37 | 2.95 | |

5 | 3.34 | 2.92 | |

| |||

| 2 | 3.76 | 3.29 |

3 | 3.69 | 3.21 | |

4 | 3.63 | 3.17 | |

5 | 3.59 | 3.13 | |

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| 2 | 3.91 | 3.36 |

3 | 3.84 | 3.28 | |

4 | 3.79 | 3.24 | |

5 | 3.74 | 3.20 |

This represents the experimental and UrQMD simulated values of generalized fractal dimension ^{16}O-AgBr, ^{28}Si-AgBr, and ^{32}S-AgBr interactions at 4.5AGeV/c.

Interactions | Order | | |
---|---|---|---|

| 2 | .892 | .826 |

3 | .877 | .810 | |

4 | .866 | .799 | |

5 | .858 | .791 | |

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| 2 | .897 | .829 |

3 | .880 | .808 | |

4 | .866 | .798 | |

5 | .857 | .788 | |

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| 2 | .924 | .836 |

3 | .907 | .815 | |

4 | .896 | .805 | |

5 | .884 | .796 |

It represents the variation of Renyi entropy 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.

It 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.

Now it will be interesting to see how the analysis will look like if one uses Shannon entropy instead of Renyi entropy. However, Shannon entropy cannot be calculated for different orders and hence we can only calculate the values of information fractal dimension for ^{16}O-AgBr, ^{28}Si-AgBr, and ^{32}S-AgBr interactions. We have calculated the values of Shannon entropies derived from the concept of Gibbs-Boltzmann theories of entropy and tabulated the values in Table ^{16}O-AgBr, ^{28}Si-AgBr, and ^{32}S-AgBr interactions are little higher than those of Renyi entropies.

This 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.

| Shannon Entropy S | Information dimension | Shannon Entropy S | Information dimension |
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| 3.59 | .922±.001 | 3.19 | .864±.002 |

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| 3.87 | .924±.002 | 3.44 | .866±.003 |

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| 4.00 | .924±.002 | 3.50 | .870±.003 |

From the values of the generalized fractal dimension calculated from Renyi entropy values we have evaluated the values of anomalous fractal dimension and hence the ratio of ^{16}O-AgBr, ^{28}Si-AgBr, and ^{32}S-AgBr interactions. It is worthwhile to point out that the spiky structure of density distribution of shower particles can also be investigated with the help of a set of bunching parameters [^{16}O-AgBr, ^{28}Si-AgBr, and ^{32}S-AgBr interactions. The calculated values of

This represents the experimental and UrQMD simulated values of ^{16}O-AgBr, ^{28}Si-AgBr, and ^{32}S-AgBr interactions at 4.5AGeV/c.

Interactions | Order | | | | |
---|---|---|---|---|---|

| 2 | 1.00 | 1.00 | 1.00 | 1.00 |

3 | 1.14 | 2.28 | 1.09 | 2.18 | |

4 | 1.24 | 3.72 | 1.15 | 3.45 | |

5 | 1.31 | 5.24 | 1.20 | 4.80 | |

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| 2 | 1.00 | 1.00 | 1.00 | 1.00 |

3 | 1.16 | 2.32 | 1.12 | 2.24 | |

4 | 1.30 | 3.90 | 1.18 | 3.54 | |

5 | 1.39 | 5.56 | 1.24 | 4.96 | |

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| 2 | 1.00 | 1.00 | 1.00 | 1.00 |

3 | 1.22 | 2.44 | 1.13 | 2.26 | |

4 | 1.37 | 4.11 | 1.19 | 3.57 | |

5 | 1.53 | 6.12 | 1.24 | 4.96 |

This represents the

Interactions | | |
---|---|---|

Experimental | UrQMD | |

| .206 | .132 |

| ||

| .262 | .156 |

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| .348 | .156 |

It represents the variation of ^{16}O-AgBr, ^{28}Si-AgBr, and ^{32}S-AgBr interactions at 4.5AGeV/c in case of experimental and UrQMD simulated events.

R.C Hwa suggested that [

We have calculated the values of ^{16}O-AgBr, ^{28}Si-AgBr, and ^{32}S-AgBr interactions. In order to search for the Ginzburg-Landau second-order phase transition we have studied the variation of ^{16}O-AgBr, ^{28}Si-AgBr, and ^{32}S-AgBr interactions at 4.5 AGeV/c for the experimental data. The variations of ^{16}O-AgBr, ^{28}Si-AgBr, and ^{32}S-AgBr interactions for experimental events. Table ^{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.

This represents the values of ^{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.

Interactions | data | |
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^{16}O-AgBr interaction at 4.5 AGeV/c | Experimental | 1.197 ± 0.003 |

UrQMD | 1.136 ± 0.004 | |

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^{28}Si-AgBr interaction at 4.5 AGeV/c | Experimental | 1.248 ± 0.010 |

UrQMD | 1.152 ± 0.007 | |

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^{32}S-AgBr interaction at | Experimental | 1.327 ± 0.022 |

UrQMD | 1.144 ± 0.007 |

It represents the fitting of ^{16}O-AgBr interactions for the experimental and UrQMD simulated data.

It represents the fitting of ^{28}Si-AgBr interactions for the experimental and UrQMD simulated data.

It represents the fitting of ^{32}S-AgBr interactions for the experimental and UrQMD simulated 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 [

We have studied the variation of ^{16}O-AgBr, ^{28}Si-AgBr, and ^{32}S-AgBr interactions in Figure

This 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.

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^{16}O-AgBrv | .116 | .119 |

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^{28}Si-AgBr | .138 | .138 |

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^{32}S-AgBr | .133 | .136 |

It represents the variation of ^{16}O-AgBr, ^{28}Si-AgBr, and ^{32}S-AgBr interactions at 4.5AGeV/c in case of experimental and UrQMD simulated events.

The experimental results have been compared with those obtained by analyzing events generated by the Ultra-Relativistic Quantum Molecular Dynamics (UrQMD) model. UrQMD is a hadronic transport model and this model can be used in the entire available range of energies to simulate nucleus-nucleus interactions. For more details about this model, readers are requested to consult [^{16}O-AgBr, ^{28}Si-AgBr, and ^{32}S-AgBr interactions at 4.5 AGeV/c. As described in our previous papers [^{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 [^{16}O-AgBr, ^{28}Si-AgBr, and ^{32}S-AgBr interactions at 4.5AGeV/c have been presented in Table ^{16}O-AgBr, ^{28}Si-AgBr, and ^{32}S-AgBr interactions along with the experimental plots. We have calculated the values of Shannon entropies and information dimension for the UrQMD data set of ^{16}O-AgBr, ^{28}Si-AgBr, and ^{32}S-AgBr interactions. The calculated values of Shannon entropy and information dimension have been presented in Table ^{16}O-AgBr, ^{28}Si-AgBr, and ^{32}S-AgBr interactions are lower than the corresponding experimental values.

As in the case of experimental data in case of UrQMD simulated data also we have calculated the values of ^{16}O-AgBr, ^{28}Si-AgBr, and ^{32}S-AgBr interactions. From the slope of the plot of ^{16}O-AgBr, ^{28}Si-AgBr, and ^{32}S-AgBr interactions. The variations of ^{16}O-AgBr, ^{28}Si-AgBr, and ^{32}S-AgBr interactions, respectively, at 4.5 AGeV/c along with the experimental plot. The variations of ^{16}O-AgBr, ^{28}Si-AgBr, and ^{32}S-AgBr interactions for the experimentally simulated events. From Table

We have studied the variation 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 has been evaluated and presented in Table

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}S-AgBr interactions at 4.5AGeV/c. Experimental results have been compared with the prediction of Ultra-Relativistic Quantum Molecular Dynamics (UrQMD) model. Qualitative information about the multifractal dynamics of particle production process has been extracted and reported in the present analysis. The significant conclusions of this analysis are as follows:

Renyi entropy values of all the interactions decrease with the order number q for both experimental and UrQMD simulated data. Renyi entropy values for UrQMD data are less than the corresponding experimental values.

Generalized fractal dimension calculated from Renyi entropy for both experimental and UrQMD simulated data decreases with the increase of order q suggesting the presence of multifractality in multiparticle production process.

The values of Shannon entropy for ^{16}O-AgBr, ^{28}Si-AgBr, and ^{32}S-AgBr interactions are little higher than those of Renyi entropies.

Degree of multifractality is found to be higher for the experimental data in comparison to the simulated data and it increases with the increase of projectile mass for the experimental data.

The experimental values of the critical exponents for ^{16}O-AgBr and ^{28}Si-AgBr interactions are lower than the critical value 1.304 required for a quark-hadron phase transition to occur while for ^{32}S-AgBr interaction the experimentally obtained values of critical exponent are higher than the critical value 1.304 signifying the absence of quark-hadron phase transition. Absence of quark-hadron phase transition is prominent for the simulated events also.

The calculated values of critical exponents obtained from our analysis increase with the increase of projectile mass for the experimental data. UrQMD predicted values of the critical exponent

Multifractal specific heat for the simulated data agrees well with the experimental data. Constancy of multifractal specific heat is reflected from our analysis.

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.

The data used to support the findings of this study are available from the corresponding author upon request.

There are no conflicts of interest in publishing the paper

The authors are grateful to Professor Pavel Zarubin of the Joint Institute of Nuclear Research (JINR), Dubna, Russia, for providing them the required emulsion data. Dr. Bhattacharyya acknowledges Professor Dipak Ghosh, Department of Physics, Jadavpur University, and Professor Argha Deb, Department of Physics, Jadavpur University, for their inspiration in the preparation of this manuscript.

^{32}S–Ag/Br interaction at 200 A GeV/c

^{+}p and

^{+}p Collisions at √s = 22 GeV

^{−2}4 seconds: Entropy production in relativistic heavy-ion collisions