Skip to main content
SpringerLink
Log in

Clan-Model of Particle Production Process-Revisited in Chaos-based Complex Network Scenario

  • Physics of Elementary Particles and Atomic Nuclei. Experiment
  • Published:
Physics of Particles and Nuclei Letters Aims and scope Submit manuscript

Abstract

The multiplicity-distribution of High-Energy-Interaction, had earlier been analysed w.r.t. clan-model and Negative-Binomial-Distribution (NBD), which described the underlying particle production process by means of cascading mechanism. Clan was defined to contain the particles stemming from the same ancestor [1] and Average-multiplicity within clan was deduced from the best fitted NBD for the multiplicity-distribution of full-phase-space. Since latest data couldn’t be confronted with NBD alone to deduce number of clans and their internal dynamics, we propose a new rigorous method using complex-network and chaos-based Visibility-Graph-algorithm to extract clusters from different rapidity-regions around the central-rapidity-(cr) of exemplary data of 32S-AgBr(200 A GeV)-interaction. These clusters are found to be scale-free and self-similar. For each cluster Power-of-Scale-freeness-of-Visibility-Graph (PSVG) and two important topological parameters: Average-clustering-co-efficient, Average-degree are extracted based on visibility of the nodes from each other in the Visibility Graph constructed from the cluster. The clan-model is revisited by correlating clusters with clans and Average-degree of the clusters with the number of particles in the clan. For each rapidity-region around (cr) the number of clusters/clans having higher values of both Average-clustering-co-efficient and PSVG, is extracted. For those clusters (Average-degree)/(number-of-particles-in-the-clan) has been calculated. It has been found that there are fewer number of clusters/clans having higher Average- clustering-co-efficient and PSVG, in the rapidity-region nearest to the (cr) and this count increases monotonically across increasing overlapping rapidity-regions around (cr)

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. A. Giovannini and L. Hove, “Negative binomial multiplicity distributions in high energyhadron collisions,” Zeitschr. Phys. C 30, 391–400 (1986). http://link.springer.com/10.1007/BF01557602.

    ADS  Google Scholar 

  2. I. Dremin and J. Gary, “Hadron multiplicities,” Phys. Rep. 349, 301–393 (2001). http://linkinghub.elsevier.com/retrieve/pii/S0370157300001174.

    Article  ADS  MATH  Google Scholar 

  3. A. Giovannini and R. Ugoccioni, “Clan structure analysis and QCD parton showers in multiparticle dynamics: an intriguing dialog between theory and experiment,” Int. J. Mod. Phys. A 20, 3897–3999 (2005). http://www.worldscientific.com/doi/abs/10.1142/S0217751X05022858.

    Article  ADS  MATH  Google Scholar 

  4. W. Kittel and E. A. de Wolf, Soft Multihadron Dynamics (World Scientific, Singapore, 2005). http://www.worldscientific.com/worldscibooks/10.1142/5805.

    Book  Google Scholar 

  5. J. F. Grosse-Oetringhaus and K. Reygers, “Charged-particle multiplicity in proton-proton collisions,” J. Phys. G: Nucl. Part. Phys. 37, 083001 (2010). http://stacks.iop.org/0954-3899/37/i=8/a=083001?key=crossref.5049962c1f37ed8f03d407d7a0edc736.

    Article  ADS  Google Scholar 

  6. M. Greenwood and G. U. Yule, “An inquiry into the nature of frequency distributions representative of multiple happenings with particular reference to the occurrence of multiple attacks of disease or of repeated accidents,” 83, 255–279 (1920).

    Google Scholar 

  7. C. Vignat and A. Plastino, “Estimation in a fluctuating medium and power-law distributions,” Phys. Lett. A 360, 415–418 (2007). http://linkinghub.elsevier.com/retrieve/pii/S037596010601084X.

    Article  ADS  Google Scholar 

  8. G. Wilk and Z. Wlodarczyk, “Power laws in elementary and heavy-ion collisions,” Eur. Phys. J. A 40, 299 (2009). http://link.springer.com/10.1140/epja/i2009-10803-9.

    Article  ADS  Google Scholar 

  9. B. Carazza and A. Gandol, “Information theory and multiplicity distributions at highenergies,” Lett. Nuovo Cim., Ser. 2 15, 553–559 (1976). http://link.springer.com/10.1007/BF02725913.

    Article  Google Scholar 

  10. A. Giovannini, “On a statistical generalization of the multiperipheral bootstrap,” Nuovo Cimento A 10, 713–722 (1972). http://link.springer.com/10.1007/BF02899771.

    Article  ADS  Google Scholar 

  11. V. V. Abramov et al., “High pT deuteron and anti-deuteron production in pp and pp a collisions at 70-GeV,” Sov. J. Nucl. Phys. 45, 845 (1987).

    Google Scholar 

  12. G. J. H. Burgers, C. Fuglesang, R. Hagedorn, and V. Kuvshinov, “Multiplicity distributionsin hadron interactions derived from the statistical bootstrap model,” Zeitschr. Phys. C 46, 465–480 (1990). http://link.springer.com/10.1007/BF01621036

    Google Scholar 

  13. S. G. Matinyan and E. B. Prokhorenko, “Branching processes and multiparticle production,” Phys. Rev. D 48, 5127–5132 (1993). https://link.aps.org/doi/10.1103/PhysRevD.48.5127

    Article  ADS  Google Scholar 

  14. S. Lee and A. Mekjian, “Development of particle multiplicity distributions using a general form of the grand canonical partition function,” Nucl. Phys. A 730, 514–547 (2004). http://linkinghub.elsevier.com/retrieve/pii/S0375947403018451

    Article  ADS  Google Scholar 

  15. A. Z. Mekjian, “Fluctuations in the statistical model of relativistic heavy ion collisions,” Nucl. Phys. A 761, 132–148 (2005). http://linkinghub.elsevier.com/retrieve/pii/S0375947405008596

    Article  ADS  Google Scholar 

  16. A. Giovannini and L. van Hove, “Negative binomial properties and clan structure in multiplicity distributions,” Acta Phys. Polon. B 19, 495 (1988).

    Google Scholar 

  17. G. Wilk and Z. Wodarczyk, “How to retrieve additional information from the multiplicity distributions,” J. Phys. G 44, 015002 (2017). http://stacks.iop.org/0954-3899/44/i=1/a=015002?key=crossref.ebd6cc0c6d47-ca04f0a270b078aea90a

    Article  ADS  Google Scholar 

  18. A. Bialas and R. Peschanski, “Moments of rapidity distributions as a measure of short-range fluctuations in highenergy collisions,” Nucl. Phys., Sect. B 273, 703–718 (1986). http://linkinghub.elsevier.com/retrieve/pii/055032138690386X

    Article  ADS  Google Scholar 

  19. G. Paladin and A. Vulpiani, “Anomalous scaling laws in multifractal objects,” Phys. Rep. 156, 147–225 (1987). http://linkinghub.elsevier.com/retrieve/pii/0370157387901104

    Article  ADS  MathSciNet  Google Scholar 

  20. E. de Wolf, I. Dremin, and W. Kittel, “Scaling laws for density correlations and fluctuations in multiparticle dynamics,” Phys. Rep. 270, 1–141 (1996). http://dx.doi.org/doi10.1016/0370-1573(95)00069-0

    Article  ADS  Google Scholar 

  21. A. Bialas and R. Peschanski, “Intermittency in multiparticle production at high energy,” Nucl. Phys., Sect. B 308, 857–867 (1988). http://linkinghub.elsevier.com/retrieve/pii/0550321388901319

    Article  ADS  Google Scholar 

  22. P. Grassberger, “Dimensions and entropies of strange attractors from a fluctuating dynamics approach,” Phys. D (Amsterdam, Neth.) 13, 34–54 (1984).

    Article  ADS  MathSciNet  MATH  Google Scholar 

  23. M. J. Halsey, L. Kadanoff, I. Procaccia, and B. Shriman, “Fractal measures and their singularities the characterization of strange sets,” Phys. Rev. A 33, 1141–1151 (1986).

    Article  ADS  MathSciNet  MATH  Google Scholar 

  24. R. Hwa, “Fractal measure in multiparticle production,” Phys. Rev. D 41, 1456–1462 (1990).

    Article  ADS  Google Scholar 

  25. F. Takagi, “Multifractal structure of multiplicity distribution in particle collisions at high energies,” Phys. Rev. Lett. 72, 32–35 (1994).

    Article  ADS  Google Scholar 

  26. D. Ghosh, A. Deb, and M. Lahiri, “Factorial and fractal analysis of the multipion production process at 350 GeV/c,” Phys. Rev D 51, 3298–3304 (1995). https://link.aps.org/doi/10.1103/PhysRevD.51.3298

    Article  ADS  Google Scholar 

  27. D. Ghosh, A. Deb, S. Bhattacharyya, and U. Datta, “Multiplicity scaling of target protons in high-energy nucleus nucleus and hadron nucleus interactions,” J. Phys. G 39, 035101 (2012). http://stacks.iop.org/0954-3899/39/i=3/a=035101?key=crossref.31779be72ac222835033a252fb863dbd

    Article  ADS  Google Scholar 

  28. C. K. Peng, S. V. Buldyrev, S. Havlin, M. Simons, H. E. Stanley, and A. L. Goldberger, “Mosaic organization of DNA nucleotides,” Phys. Rev. E 49, 1685–1689 (1994). http://link.aps.org/doi/10.1103/Phys-RevE.49.1685

    Article  ADS  Google Scholar 

  29. J. W. Kantelhardta, E. Koscielny-Bundea, H. H. A. Rego, S. Havlinb, and A. Bundea, “Detecting long-range correlations with detrended fluctuation analysis,” Phys. A (Amsterdam, Neth.) 295, 441–454 (2001).

    Article  ADS  Google Scholar 

  30. M. S. Taqqu and V. W. W. Teverovsky, “Estimations for long-range dependence: an empiricalstudy,” Fractals 3, 785–788 (1995).

    Article  MATH  Google Scholar 

  31. Z. P. I. Chen, K. Hu, and H. Stanley, “Effect of nonstationarities on detrended fluctuation analysis,” Phys. Rev. E 65, 041107–041122 (2002).

    Article  ADS  Google Scholar 

  32. J. W. Kantelhardt, S. A. Zschiegner, E. Koscielny-Bunde, A. Bunde, S. Havlin, and H. E. Stanley, “Multifractal detrended fluctuation analysis of nonstationary time series,” Phys. A (Amsterdam, Neth.) 02, 01383–3 (2002).

    MATH  Google Scholar 

  33. W. Y. Zhang and C. Y. Qian, “Multifractal structure of pseudorapidity and azimuthal distributions of the shower particles in Au + Au collisions,” J. Mod. Phys. A 18, 2809–2816 (2007).

    Google Scholar 

  34. C. Albajar, O. C. Allkofer, R. J. Apsimon, et al., “Multifractal analysis of minimum bias events in s**(1/2) = 630-GeV anti-p p collisions,” Zeitschr. Phys. C 56, 37–46 (1992). http://link.springer.com/10.1007/BF01589705

    ADS  Google Scholar 

  35. D. Ghosh, A. Deb, P. Bandyopadhyay, M. Mondal, S. Bhattacharyya, J. Ghosh, and K. Patra Kumar, “Evidence of multifractality and constant specific heat in hadronic collisions at high energies,” Phys. Rev. C 65, 067902 (2002). https://link.aps.org/doi/10.1103/Phys-RevC.65.067902

    Article  ADS  Google Scholar 

  36. M. K. Suleymanov, M. Sumbera, and I. Zborovsky, “Entropy and multifractal analysis of multiplicity distributions from pp simulated events up to LHC energies,” (2003).

    Google Scholar 

  37. D. Ghosh, A. Deb, S. Pal, P. K. Haldar, S. Bhattacharyya, P. Mandal, S. Biswas, and M. Mondal, “Evidence of fractal behavior of pions and protons in high energy interactions an experimental investigation,” Fractals 13, 325–339 (2005). http://www.worldscientific.com/doi/abs/10.1142/S0218348X05002921

    Article  Google Scholar 

  38. E. G. Ferreiro and C. Pajares, “High multiplicity pp events and J/nps in production at LHC,” (2012).

    Google Scholar 

  39. P. Mali, S. Sarkar, S. Ghosh, A. Mukhopadhyay, and G. Singh, “Multifractal detrended fluctuation analysis of particle density fluctuations in high-energy nuclear collisions,” Phys. A (Amsterdam, Neth) 424, 25–33 (2015). http://linkinghub.elsevier.com/retrieve/pii/S0378437114010796

    Article  Google Scholar 

  40. D. Ghosh, S. Dutta, and S. Chakraborty, “Multifractal detrended cross-correlation analysis for epileptic patient in seizure and seizure free status,” Chaos, Solitons Fractals 67, 1–10 (2014).

    Article  ADS  MathSciNet  Google Scholar 

  41. A. Bhaduri, S. Bhaduri, and D. Ghosh, “Visibility graph analysis of heart rate time series and bio-marker of congestive heart failure,” Phys. A (Amsterdam, Neth) 482, 786–795 (2017). http://linkinghub.elsevier.com/retrieve/pii/S0378437117303990

    Article  ADS  Google Scholar 

  42. S. Bhaduri and D. Ghosh, “Speech, music and multifractality,” Curr. Sci. 110, 1817–1822 (2016). http://www.currentscience.ac.in/Volumes/110/9/1822.pdf

    Article  Google Scholar 

  43. S. Bhaduri, R. Das, and D. Ghosh, “Non-invasive detection of Alzheimer’s disease-multifractality of emotional speech,” J. Neurol. Neurosci. 7 (2:84), 1–7 (2016). http://www.jneuro.com/neurology-neuroscience/noninvasive-detection-of-alzheimers-disease-multifractality-of-emotional-speech.phpid=9120.

  44. F. Wang, G. p. Liao, X. y. Zhou, and W. Shi, “Multifractal detrended cross-correlation analysis for power markets,” Nonlin. Dyn. 72, 353–363 (2013).

    Article  Google Scholar 

  45. S. Bhaduri, A. Bhaduri, and D. Ghosh, “A new approach of chaos and complex network method to study fluctuation and phase transition in nuclear collision at high energy,” Eur. Phys. J. A 53 (6), 135 (2017). http://dx.doi.org/doi10.1140/epja/i2017-12332-4

    Article  ADS  Google Scholar 

  46. R. Albert and A. L. Barabási, “Statistical mechanics of complex networks,” Rev. Mod. Phys. 74, 47–97 (2002). http://link.aps.org/doi/10.1103/RevModPhys.74.47

    Article  ADS  MathSciNet  MATH  Google Scholar 

  47. A. L. Barabási, “The network takeover,” Nat. Phys. 8, 14–16 (2011). http://dx.doi.org/doi10.1038/nphys2188

    Google Scholar 

  48. S. Havlin, D. Y. Kenett, E. Ben-Jacob, A. Bunde, R. Cohen, H. Hermann, J. W. Kantelhardt, J. Kertész, S. Kirkpatrick, J. Kurths, J. Portugali, and S. Solomon, “Challenges in network science: applications to infrastructures, climate, social systems and economics,” Eur. Phys. J. Spec. Top. 214, 273–293 (2012). http://www.springerlink.com/index/10.1140/epjst/e2012-01695-x

    Article  Google Scholar 

  49. L. Zhao, W. Li, and X. Cai, “Structure and dynamics of stock market in times of crisis,” Phys. Lett. A 380, 654–666 (2016). http://linkinghub.elsevier.com/retrieve/pii/S0375960115009937

    Article  ADS  MathSciNet  Google Scholar 

  50. L. Lacasa, B. Luque, F. Ballesteros, J. Luque, and J. C. Nuno, “From time series to complex networks: The visibility graph,” Proc. Natl. Acad. Sci. U.S.A. 105, 4972–4975 (2008). http://www.pnas.org/cgi/doi/10.1073/pnas.0709247105

    Article  ADS  MathSciNet  MATH  Google Scholar 

  51. L. Lacasa, B. Luque, J. Luque, and J. C. Nuno, “The visibility graph: a new method for estimating the Hurst exponent of fractional Brownian motion,” Europhys. Lett. 86 (3) (2009).

    Google Scholar 

  52. D. Hofstadter, “Godel, Escher, Bach: an eternal golden braid,” New Society 54, 431 (1980).

    MATH  Google Scholar 

  53. M. A. Stern, “Über eine zahlentheoretische Funktion,” J. Reine Angew. Math. 55, 193–220 (1858). http://www.digizeitschriften.de/dms/img/?PPN=GDZPPN002150301

    Article  MathSciNet  Google Scholar 

  54. M. Schroeder and K. Wiesenfeld, “Fractals, chaos, power laws: minutes from an infinite paradise,” Phys. Today 44, 91 (1991). http://physicstoday.scitation.org/doi/10.1063/1.2810323

    Article  MATH  Google Scholar 

  55. J. M. Hausdorff, P. L. Purdon, C. K. Peng, Z. Ladin, J. Y. Wei, and A. L. Goldberger, “Fractal dynamics of human gait: stability of long-range correlations in stride interval fluctuations,” J. Appl. Physiol. 80, 1448–1457 (1996). http://jap.physiology.org/content/80/5/1448.short

    Article  Google Scholar 

  56. A. L. Goldberger, L. A. N. Amaral, J. M. Hausdorff, P. C. Ivanov, C. K. Peng, and H. E. Stanley, “Fractal dynamics in physiology: alterations with disease and aging,” Proc. Nat. Acad. Sci. 99, 2466–2472 (2002).

    Article  ADS  Google Scholar 

  57. S. Jiang, C. Bian, X. Ning, and Q. D. Y. Ma, “Visibility graph analysis on heartbeat dynamics of meditation training,” Appl. Phys. Lett. 102, 253–702 (2013).

    Google Scholar 

  58. S. Bhaduri and D. Ghosh, “Electroencephalographic data analysis with visibility graph technique for quantitative assessment of brain dysfunction,” Clin. EEG Neurosci., 3–8 (2014). http://www.ncbi.nlm.nih.gov/pubmed/24781371

    Google Scholar 

  59. A. Bhaduri and D. Ghosh, “Quantitative assessment of heart rate dynamics during meditation: an ECG based study with multi-fractality and visibility graph,” Front. Physiol. 7 (2016). http://journal.frontiersin.org/Article/10.3389/fphys.2016.00044/abstract

    Google Scholar 

  60. P. Nilanjana, B. Anirban, B. Susmita, and G. Dipak, “Non-invasive alarm generation for sudden cardiac arrest: a pilot study with visibility graph technique,” Transl. Biomed. 7 (3) (2016). http://www.transbiomedicine.com/translational-biomedicine/noninvasive-alarmgeneration-for-sudden-cardiac-arrest-a-pilot-study-withvisibility-graph-technique.phpid=13601.

    Google Scholar 

  61. S. Bhaduri, A. Chakraborty, and D. Ghosh, “Speech emotion quantification with chaos-based modified visibility graph-possible precursor of suicidal tendency,” J. Neurol. Neurosci. 7 (3) (2016). http://www.jneuro.com/neurology-neuroscience/speech-emotion-quantification-with-chaosbased-modified-visibility-graph-possible-precursor-of-suicidal-tendency.phpid=9457.

    Google Scholar 

  62. S. Bhaduri and D. Ghosh, “Multiplicity fluctuation and phase transition in high-energy collision. A chaosbased study with complex network perspective,” Int. J. Mod. Phys. A 31, 1650185 (2016). http://www.worldscientific.com/doi/abs/10.1142/S0217751X16501852

    Article  ADS  Google Scholar 

  63. S. Bhaduri and D. Ghosh, “Pion fluctuation in highenergy collisions–a chaos-based quantitative estimation with visibility graph technique,” Acta Phys. Polon. B 48, 741 (2017). http://www.actaphys.uj.edu.pl/findarticle?series=Reg&vol=48&page=741

    Article  ADS  Google Scholar 

  64. S. Bhaduri and D. Ghosh, “Fractal study of pion void probability distribution in ultrarelativistic nuclear collision and its target dependence,” Mod. Phys. Lett. A 31, 1650158 (2016). http://www.worldscientific.com/doi/10.1142/S0217732316501583

    Article  ADS  Google Scholar 

  65. A. Bhaduri, S. Bhaduri, and D. Ghosh, “Azimuthal pion fluctuation in ultra relativistic nuclear collisions and centrality dependence–a study with chaos based complex network analysis,” Phys. Part. Nucl. Lett. 14, 576–583 (2017). http://dx.doi.org/doi10.1134/S1547477117040033

    Article  Google Scholar 

  66. M. Ahmadlou, H. Adeli, and A. Adeli, “Improved visibility graph fractality with application for the diagnosis of Autism Spectrum Disorder,” Phys. A (Amsterdam, Neth.) 391, 4720–4726 (2012).

    Article  ADS  Google Scholar 

  67. E. Estrada, “Quantifying network heterogeneity,” Phys. Rev. E 82, 066102 (2010). http://link.aps.org/doi/10.1103/PhysRevE.82.066102

    Article  ADS  Google Scholar 

  68. D. J. J. Watts and S. H. H. Strogatz, “Collective dynamics of “small-world” networks,” Nature (London, U.K.) 393, 440–442 (1998).

    Article  ADS  MATH  Google Scholar 

  69. S. C. Carlson, Graph Theory (2014). http://www.britannica.com/EBchecked/topic/242012/graph-theory

    Google Scholar 

  70. D. B. Johnson, “Efficient algorithms for shortest paths in sparse networks,” J. ACM 24, 1–13 (1977).

    Article  MathSciNet  MATH  Google Scholar 

  71. M. E. J. Newman, “Assortative mixing in networks,” Phys. Rev. Lett. 89, 208701 (2002). http://link.aps.org/doi/10.1103/PhysRevLett.89.208701

    Article  ADS  Google Scholar 

  72. L. Adamczyk, J. K. Adkins, G. Agakishiev, et al., “Beam energy dependence of moments of the net-charge multiplicity distributions in Au+Au collisions at RHIC,” Phys. Rev. Lett. 113, 092301 (2014). https://link.aps.org/doi/10.1103/PhysRevLett.113.092301

    Article  ADS  Google Scholar 

  73. M. Ester, H. P. Kriegel, J. Sander, and X. Xu, “A density-based algorithm for discovering clusters in large spatial databases with noise,” in Proceedings of the 2nd International Conference on Knowledge Discovery and Data Mining, 1996, pp. 226–231. https://www.aaai.org/Papers/KDD/1996/KDD96-037.pdf

    Google Scholar 

  74. D. Ghosh, A. Deb, S. Bhattacharyya, and U. Datta, “Rapidity dependence of multiplicity fluctuations and correlations in high-energy nucleus-nucleus interactions,” Pramana J. Phys. 77, 297–313 (2011). http://link.springer.com/10.1007/s12043-011-0131-2

    Article  ADS  Google Scholar 

  75. C. F. P. H. F. Powell, and D. H. Perkins, The Study of Elementary Particles by the Photographic Method (Pergamon, Oxford, 1959).

    Google Scholar 

  76. A. Clauset, C. R. Shalizi, and M. E. J. Newman, “Power-law distributions in empirical data,” SIAM Rev. 51, 661–703 (2009). http://epubs.siam.org/doi/abs/10.1137/070710111

    Article  ADS  MathSciNet  MATH  Google Scholar 

  77. J. L. Devore, Probability and Statistics for Engineering and the Sciences (2015). https://books.google.com.mx/booksid=zzV-BAAAQBAJ.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Deepa Ghosh Research Foundation, Kolkata, 700031, India

    S. Bhaduri, A. Bhaduri & D. Ghosh

Authors
  1. S. Bhaduri
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. A. Bhaduri
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. D. Ghosh
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to S. Bhaduri.

Additional information

The article is published in the original.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Bhaduri, S., Bhaduri, A. & Ghosh, D. Clan-Model of Particle Production Process-Revisited in Chaos-based Complex Network Scenario. Phys. Part. Nuclei Lett. 15, 446–455 (2018). https://doi.org/10.1134/S1547477118040040

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S1547477118040040

Search

Navigation