Takagi Moment Research Articles

Constant Specific Heat Approximation in Multifractal Thermodynamics in Multiparticle Production in Relativistic Heavy-Ion Collisions

The study of specific heat is motivated by the fact that a sudden change in the value of specific heat might be interpreted as a signal of a phase transition. It is also an established fact that multifractal analysis has been proved to be highly effective in characterizing fluctuations which is considered to be important tool for understanding the mechanism of quark-gluon plasma (QGP) in high energy nucleus-nucleus collisions. The Present research work provides some fascinating investigations on multifractal specific heat, c using the concept of entropy, f_q which is found as a potential procedure in the study of multifractal specific heat along with the earlier known approaches. The investigations are done for the produced shower particles in nuclear emulsion detector for ²⁸Si-nucleus interactions at 14.5 A GeV/c in the framework of generalized dimension. An attempt is also made to discuss certain universal properties of multifractal specific heat and entropy. We have computed c, by applying the methodology of modified and Takagi moments (T_q). Experimental results are compared with the predictions of LUND model FRITIOF. Moreover, the constant-specific heat method, which is based on the concept of entropy and is commonly accepted in conventional thermodynamics, is demonstrated to be suitable in multifractal thermodynamics also. The values of ‘c’ calculated from these methods are compared with constant specific heat approximations (CSHs) obtained using multifractal entropy (f_q). It is found that the values of ‘c’, estimated using Takagi approach are consistent with those of Bershadskii's work as compared to those calculated using (G_a^m) moments and multifractal entropy, f_q. This is obtained for both experimental and for the FRITIOF generated data for the three types of interactions namely, CNO, emulsion and AgBr. The findings of this paper reveal useful information regarding the choice of method used and our results are consistent with CSH approximation for both experimental and simulated data and also in conjunction with recent studies on multifractal specific heat.

A STUDY OF MULTIFRACTALITY AND PHASE TRANSITION IN HEAVY-ION COLLISIONS — EXPERIMENTAL DATA VERSUS MODEL SIMULATION

In this paper, we have investigated the presence of multifractality and the possibility of quark–hadron phase transition in the multiparticle production in [Formula: see text]O–AgBr, [Formula: see text]Si–AgBr and [Formula: see text]S–AgBr interactions at 4.5[Formula: see text]AGeV/[Formula: see text] in the framework of Ginzburg–Landau theory of second-order phase transition following the Takagi moment methodology. The same analysis with the factorial moment method has also been performed. Experimental results have been compared with the prediction of UrQMD and AMPT models. A comparison of experimental results with the analysis of UrQMD simulated RHIC data by the method of Takagi moment has also been presented. Comparison of experimental results with previous results of emulsion experiments has been mentioned.

Study of phase transition and its dependence on target excitation using fractal properties of pionization process

In the present work, phase transition and its dependence on target excitation in terms of Levy index and Ginzburg–Landau formalism using fractal properties of pionization process have been studied in 16O–AgBr and 32S–AgBr interactions at incident energies of 60 and 200 AGeV, respectively. For this purpose, the pion data are divided into three sets depending upon the number of grey tracks (ng). Two multifractal moments, namely, G-moment and Takagi moment, have been used as analyzing tools. The study reveals that the pion density distribution possesses multifractal structure for the two types of interactions and at all degrees of target excitation. Different types of phase transitions (thermal and non-thermal) at different energies are indicated by the data but no signal of quark–gluon plasma (QGP) phase transition is obtained.

Multifractal study and multifractal specific heat of singly charged particles produced in 32S–Em interactions at 200 AGeV

Takagi moment method has been applied to investigate the multifractal behaviour of charged pions produced in the interaction of 32S beam with nuclear emulsion at 200 AGeV in pseudo-rapidity space. Following this method the multiplicity distributions are found to obey power law behaviour for experimental data for different nuclear targets. The generalised fractal dimensions, Dq are found to decrease with increasing order of moments, q. The observation reflects the existence of multifractality in multiparticle production. The linear behaviour of Dq as a function of ln q/(q − 1) favours the multifractal Bernoulli representation and the multifractal specific heat seems to have a universal value (∼1/4) that does not depend on the type of ions and their energy. The experimental results have been compared with those obtained from simulated events of the Monte Carlo code FRITIOF.

Evidence of deconfinement phase transition in multiparticle production in relativistic nuclear collisions

This paper presents the results of an investigation pertaining to multifractal structure, degree of multifractality and occurrence of deconfinement phase transition. These results are obtained by carrying out analysis of generalized dimensions and Levy index involving relativistic charged particles produced on 14.5A GeV/c28 Si -nucleus collisions following Takagi's approach. The values of Levy index, μ, generalized fractal dimensions, Dq and degree of multifractality, βq, for different orders of moments are calculated in terms of Takagi moments for the experimental, FRITIOF, HIJING and random Monte Carlo (MC-RAND) generated events. The values of multifractal specific heats for all the above data sets are extracted using generalized dimensions, Dq. The analysis also shows small variations in the degree of multifractality for the experimental, FRITIOF and HIJING simulated events. However, the degree of multifractality is quite larger for the random Monte Carlo 28 Si -nucleus generated events at 14.5A GeV/c. Furthermore, the analyses of multifractal specific heat and Levy index in terms of Takagi moments support the presence of multifractality and occurrence of deconfinement phase transition in the experimental and simulated data sets, but for the uncorrelated Monte Carlo simulated events, occurrence of nonthermal phase transition is revealed.

DYNAMICS OF FRACTALITY AND PHASE TRANSITION IN MULTIPARTICLE PRODUCTION IN RELATIVISTIC NUCLEUS–NUCLEUS COLLISIONS

This paper reports the results of an investigation regarding occurrence of second-order phase transition in 14.5A GeV /c28 Si -nucleus interactions using the method of scaled factorial moments. Incidentally, the value of the universal scaling exponent, ν, for our experimental data is found to be 1.224±0.068, which is quite close to its critical value 1.304. An attempt is also made to search for the evidence of phase transition in terms of Levy index, μ, using scaled factorial moments as well as Takagi moments for both experimental and FRITIOF generated data sets. Average value of μ, calculated from Fq moments, turns out to be more than unity but is less than unity when estimated in terms of Takagi moments for both the data sets. Thus the analyses carried out in terms of Fq and Takagi moments reveal the occurrence of nonthermal and thermal phase transitions, respectively.

A COMPARATIVE STUDY OF THE EXPERIMENTAL AND SIMULATED VALUES OF MULTIFRACTAL SPECIFIC HEAT IN 14.5A GeV/c HEAVY-ION COLLISION

An attempt is made to study multifractal specific heat from Fq, modified Gq and Takagi moments, which can provide some interesting information about event-by-event fluctuations in heavy-ion collisions at high energies. It is an established fact that constancy of the specific heat (CSH) of solids and gases helps explain thermodynamical behavior of intermittency and multifractality in A–A collisions at different projectile energies. It is worthmentioning that the present paper deals with multifractal specific heat, c, extracted from Fq, modified Gq and Takagi moments for the experimental as well as FRITIOF generated data on 14.5A GeV /c28 Si -nucleus interactions. The results obtained using the three approaches are compared for both the data sets. The observed power law behavior reflects self-similar property of the multiparticle production process. The common feature of the multifractal specific heat obtained using the three approaches is that there is no systematic variation in its value for different orders of the moments. Furthermore, the value of c is found to be different for both the data sets and no strong target mass dependence is observed in any of the cases. A noteworthy observation is that a non-zero finite value of the multifractal specific heat may be considered a good signal for the presence of multifractality in the relativistic charged particle multiplicity distribution.

Multifractality in charged pion production at a few GeV/n

The multifractal analysis of charged pion produced in 16O–AgBr interactions at 2.1 AGeV and 24Mg–AgBr interactions at 4.5 AGeV has been investigated using the Takagi moment method in pseudo-rapidity space. The generalized fractal dimensions Dq are determined for these two interactions for different orders of moment. Experimental data reflects multifractal geometry in the multipion production process. From the knowledge of the generalized fractal dimensions Dq, the multifractal specific heat has also been evaluated for these data. A detailed comparison is presented with the results obtained from 12C–AgBr and 28Si–AgBr interactions at 4.5 AGeV/c, 28Si–AgBr interactions at 14.5 A GeV/c, 16O–AgBr interactions 60 AGeV and 32S–AgBr interactions 200 AGeV from the recent literatures.

Levy index analysis for a multifractality and phase transition study of target fragments in ring-like and jet-like events

In the present work, Levy indices μ are measured for target fragments for both ring-like and jet-like events in 16O–AgBr interaction at 60 AGeV and 32S–AgBr interaction at 200 AGeV. The Fq moment analysis reveals unphysical results and hence the Takagi moment analysis is adopted to get an insight into the underlying physical phenomenon. The analysis indicates different degrees of multifractality in the two types of events (ring-like and jet-like) for 32S–AgBr interaction. Also, the Levy stable law is applied to calculate the Levy index μ, which indicates different rates of cascading in different types of events.

A comparative study of multifractal moments in relativistic heavy-ion collisions

A study of the fractal structure of relativistic shower particles produced in the interactions of 28Si and 12C nuclei at 4.5 A GeV/c with nuclear emulsion using the methods of Takagi moments, Tq, factorial moments, Fq, and modified multifractal moments, Gq, has been performed. The dependences of these moments on the number of bins M are found to follow power law behaviour for the experimental data in pseudorapidity phase space. The calculated values of the generalized fractal dimensions, Dq, are found to decrease with the increasing order of moments, q. This observation indicates the existence of multifractality and a self-similar cascade mechanism in multiparticle production. The multifractal Bernoulli representation Dq = D∞ + c ln q/(q − 1) is found to support the linear dependence of Dq on ln q/(q − 1). Finally, from the analysis of the data it can be concluded that no universality in the values of the multifractal specific heat, c, calculated from Tq, Fq and Gq moments is found. Some consistency seems to be observed in the values of the specific heat obtained from the Takagi method.

EVIDENCE OF FRACTAL BEHAVIOR OF PIONS AND PROTONS IN HIGH ENERGY INTERACTIONS — AN EXPERIMENTAL INVESTIGATION

Different fractal methods (Factorial moment, Takagi moment) have been used for the analysis of fractal behavior of multiparticle production in π-- AgBr interaction at 350 GeV/c of compound multiplicity distribution in emission angle (cos θ) space and azimuthal angle (ϕ) space. The generalized fractal dimensions Dq are determined according to two methods. Results indicate multifractal behavior in compound multiplicity distribution. From the knowledge of Dq, the multifractal specific heat is calculated. Results are compared with the corresponding results of shower multiplicity.

Evidence of Multifractal Structure of Compound Multiplicity Distribution in24Mg-AgBr Interactions at 4.5AGeV

This paper reports an analysis on fractality using the experimental data of compound multiplicity (pions + target protons) distribution in 24Mg-AgBr interactions at 4.5 AGeV following the method proposed by F. Takagi. Takagi moments have been calculated for 2nd, 3rd, 4th, and 5th order in the emission angle and azimuthal angle space. The results of our study reveal the multi-fractal structure of compound multiplicity distribution.

FRACTAL BEHAVIOR OF NUCLEAR FRAGMENTS IN HIGH ENERGY INTERACTIONS

The multifractal analysis of data on nuclear fragments obtained from 28 Si-AgBr interactions at 14.5 A GeV is performed using three different methods (the factorial moments, G-moments and Takagi moments). The generalized fractal dimensions Dq is determined from all these methods. Data reflects multifractal geometry for the nuclear fragments. From the knowledge of Dq, the multifractal specific heat is calculated for this data and also for 16 O-AgBr interactions at 60 A GeV and 32 S-AgBr interactions at 200 A GeV.