Skip to main content
SpringerLink
Log in

Multiplicity distributions in canonical and microcanonical statistical ensembles

  • Regular Article - Theoretical Physics
  • Published:
The European Physical Journal C Aims and scope Submit manuscript

Abstract

The aim of this paper is to introduce a new technique for the calculation of observables, in particular multiplicity distributions, in various statistical ensembles at finite volume. The method is based on Fourier analysis of the grand canonical partition function. A Taylor expansion of the generating function is used to separate contributions to the partition function in their power in volume. We employ Laplace’s asymptotic expansion to show that any equilibrium distribution of multiplicity, charge, energy, etc. tends to a multivariate normal distribution in the thermodynamic limit. A Gram–Charlier expansion additionally allows for the calculation of finite volume corrections. Analytical formulas are presented for the inclusion of resonance decay and finite acceptance effects directly into the partition function of the system. This paper consolidates and extends previously published results of the current investigation into the properties of statistical ensembles.

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. E. Fermi, Prog. Theor. Phys. 5, 570 (1950)

    Article  ADS  MathSciNet  Google Scholar 

  2. R. Hagedorn, Nucl. Phys. B 24, 93 (1970)

    Article  ADS  Google Scholar 

  3. J. Cleymans, D. Elliott, A. Keranen, E. Suhonen, Phys. Rev. C 57, 3319 (1998)

    Article  ADS  Google Scholar 

  4. J. Cleymans, H. Oeschler, K. Redlich, Phys. Rev. C 59, 1663 (1999)

    Article  ADS  Google Scholar 

  5. R. Averbeck, R. Holzmann, V. Metag, R.S. Simon, Phys. Rev. C 67, 024903 (2003)

    Article  ADS  Google Scholar 

  6. P. Braun-Munzinger, J. Stachel, J.P. Wessels, N. Xu, Phys. Lett. B 344, 43 (1995)

    Article  ADS  Google Scholar 

  7. P. Braun-Munzinger, J. Stachel, J.P. Wessels, N. Xu, Phys. Lett. B 365, 1 (1996)

    Article  ADS  Google Scholar 

  8. P. Braun-Munzinger, I. Heppe, J. Stachel, Phys. Lett. B 465, 15 (1999)

    Article  ADS  Google Scholar 

  9. F. Becattini, M. Gazdzicki, A. Keranen, J. Manninen, R. Stock, Phys. Rev. C 69, 024905 (2004)

    Article  ADS  Google Scholar 

  10. J. Adams et al. (STAR Collaboration), Nucl. Phys. A 757, 102 (2005)

    Article  ADS  Google Scholar 

  11. J. Cleymans, H. Oeschler, K. Redlich, S. Wheaton, Phys. Rev. C 73, 034905 (2006)

    Article  ADS  Google Scholar 

  12. J. Cleymans, K. Redlich, Phys. Rev. C 60, 054908 (1999)

    Article  ADS  Google Scholar 

  13. J. Cleymans, K. Redlich, Phys. Rev. Lett. 81, 5284 (1998)

    Article  ADS  Google Scholar 

  14. F. Becattini, J. Manninen, M. Gaździcki, Phys. Rev. C 73, 044905 (2006)

    Article  ADS  Google Scholar 

  15. A. Andronic, P. Braun-Munzinger, J. Stachel, Nucl. Phys. A 772, 167 (2006)

    Article  ADS  Google Scholar 

  16. F. Becattini, Z. Phys. C 69, 485 (1996)

    Article  Google Scholar 

  17. F. Becattini, U.W. Heinz, Z. Phys. C 76, 269 (1997)

    Article  Google Scholar 

  18. F. Becattini, G. Passaleva, Eur. Phys. J. C 23, 551 (2002)

    Article  ADS  Google Scholar 

  19. R. Hagedorn, CERN-TH-7190/94, 1994

  20. R. Hagedorn, CERN lectures, Thermodynamics of Strong Interaction (1970)

  21. R. Stock, Phys. Lett. B 456, 277 (1999)

    Article  ADS  Google Scholar 

  22. F. Becattini, J. Phys. Conf. Ser. 5, 175 (2005)

    Article  ADS  Google Scholar 

  23. F. Becattini, L. Ferroni, Eur. Phys. J. C 51, 899 (2007)

    Article  ADS  MathSciNet  Google Scholar 

  24. M. Gazdzicki, M.I. Gorenstein, S. Mrowczynski, Phys. Lett. B 585, 115 (2004)

    Article  ADS  Google Scholar 

  25. M.I. Gorenstein, M. Gazdzicki, O.S. Zozulya, Phys. Lett. B 585, 237 (2004)

    Article  ADS  Google Scholar 

  26. I.N. Mishustin, Phys. Rev. Lett. 82, 4779 (1999)

    Article  ADS  Google Scholar 

  27. I.N. Mishustin, Nucl. Phys. A 681, 56–63 (2001)

    Article  ADS  Google Scholar 

  28. H. Heiselberg, A.D. Jackson, Phys. Rev. C 63, 064904 (2001)

    Article  ADS  Google Scholar 

  29. M.A. Stephanov, K. Rajagopal, E.V. Shuryak, Phys. Rev. Lett. 81, 4816 (1998)

    Article  ADS  Google Scholar 

  30. M.A. Stephanov, K. Rajagopal, E.V. Shuryak, Phys. Rev. D 60, 114028 (1999)

    Article  ADS  Google Scholar 

  31. M.A. Stephanov, Acta Phys. Pol. B 35, 2939 (2004)

    ADS  Google Scholar 

  32. S. Jeon, V. Koch, Phys. Rev. Lett. 85, 2076 (2000)

    Article  ADS  Google Scholar 

  33. S. Jeon, V. Koch, Phys. Rev. Lett. 85, 2076 (2000)

    Article  ADS  Google Scholar 

  34. M. Asakawa, U.W. Heinz, B. Muller, Phys. Rev. Lett. 85, 2072 (2000)

    Article  ADS  Google Scholar 

  35. M. Asakawa, U.W. Heinz, B. Muller, Phys. Rev. Lett. 85, 2072 (2000)

    Article  ADS  Google Scholar 

  36. S.V. Afanasev et al. (NA49 Collaboration), Phys. Rev. Lett. 86, 1965 (2001)

    Article  ADS  Google Scholar 

  37. M.M. Aggarwal et al. (WA98 Collaboration), Phys. Rev. C 65, 054912 (2002)

    Article  ADS  Google Scholar 

  38. J. Adams et al. (STAR Collaboration), Phys. Rev. C 68, 044905 (2003)

    Article  ADS  Google Scholar 

  39. C. Roland et al. (NA49 Collaboration), J. Phys. G 30, S1381 (2004)

    Article  ADS  Google Scholar 

  40. Z.W. Chai et al. (PHOBOS Collaboration), J. Phys. Conf. Ser. 27, 128 (2005)

    Article  ADS  Google Scholar 

  41. M. Rybczynski et al. (NA49 Collaboration), J. Phys. Conf. Ser. 5, 74 (2005)

    Article  ADS  Google Scholar 

  42. H. Appelshauser et al. (NA49 Collaboration), Phys. Lett. B 459, 679 (1999)

    Article  ADS  Google Scholar 

  43. D. Adamova et al. (CERES Collaboration), Nucl. Phys. A 727, 97 (2003)

    Article  ADS  Google Scholar 

  44. T. Anticic et al. (NA49 Collaboration), Phys. Rev. C 70, 034902 (2004)

    Article  ADS  Google Scholar 

  45. S.S. Adler et al. (PHENIX Collaboration), Phys. Rev. Lett. 93, 092301 (2004)

    Article  ADS  Google Scholar 

  46. J. Adams et al. (STAR Collaboration), Phys. Rev. C 71, 064906 (2005)

    Article  ADS  Google Scholar 

  47. B. Lungwitz et al. (NA49 Collaboration), PoS C FRNC2006, 024 (2006)

    Google Scholar 

  48. V.V. Begun, M. Gazdzicki, M.I. Gorenstein, M. Hauer, V.P. Konchakovski, B. Lungwitz, Phys. Rev. C 76, 024902 (2007)

    Article  ADS  Google Scholar 

  49. S.V. Afanasiev et al. (The NA49 Collaboration), Phys. Rev. C 66, 054902 (2002)

    Article  ADS  Google Scholar 

  50. M. Gazdzicki et al. (NA49 Collaboration), J. Phys. G 30, S701 (2004)

    Article  ADS  Google Scholar 

  51. M. Gazdzicki et al. (NA49-future Collaboration), PoS C POD2006, 016 (2006)

    Google Scholar 

  52. C.M. Ko, V. Koch, Z.w. Lin, K. Redlich, M.A. Stephanov, X.N. Wang, Phys. Rev. Lett. 86, 5438 (2001)

    Article  ADS  Google Scholar 

  53. O. Fochler, S. Vogel, M. Bleicher, C. Greiner, P. Koch-Steinheimer, Z. Xu, Phys. Rev. C 74, 034902 (2006)

    Article  ADS  Google Scholar 

  54. G. Torrieri, J. Rafelski, Phys. Lett. B 509, 239 (2001)

    Article  ADS  Google Scholar 

  55. V.V. Begun, M. Gaździcki, M.I. Gorenstein, O.S. Zozulya, Phys. Rev. C 70, 034901 (2004)

    Article  ADS  Google Scholar 

  56. V.V. Begun, M.I. Gorenstein, A.P. Kostyuk, O.S. Zozulya, Phys. Rev. C 71, 054904 (2005)

    Article  ADS  Google Scholar 

  57. V.V. Begun, M.I. Gorenstein, Phys. Lett. B 653, 190 (2007)

    Article  ADS  Google Scholar 

  58. V.V. Begun, M.I. Gorenstein, A.P. Kostyuk, O.S. Zozulya, J. Phys. G 32, 935 (2006)

    Article  ADS  Google Scholar 

  59. V.V. Begun, M.I. Gorenstein, M. Hauer, V.P. Konchakovski, O.S. Zozulya, Phys. Rev. C 74, 044903 (2006)

    Article  ADS  Google Scholar 

  60. V.V. Begun, M.I. Gorenstein, A.P. Kostyuk, O.S. Zozulya, Phys. Rev. C 71, 054904 (2005)

    Article  ADS  Google Scholar 

  61. V.V. Begun, M.I. Gorenstein, O.S. Zozulya, Phys. Rev. C 72, 014902 (2005)

    Article  ADS  Google Scholar 

  62. F. Becattini, A. Keränen, L. Ferroni, T. Gabbriellini, Phys. Rev. C 72, 064904 (2005)

    Article  ADS  Google Scholar 

  63. J. Cleymans, K. Redlich, L. Turko, Phys. Rev. C 71, 047902 (2005)

    Article  ADS  Google Scholar 

  64. J. Cleymans, K. Redlich, L. Turko, J. Phys. G 31, 1421 (2005)

    Article  ADS  Google Scholar 

  65. G. Torrieri, S. Jeon, J. Rafelski, Phys. Rev. C 74, 024901 (2006)

    Article  ADS  Google Scholar 

  66. G. Torrieri, J. Phys. G 32, S195 (2006)

    Article  ADS  Google Scholar 

  67. S. Jeon, V. Koch, Phys. Rev. Lett. 83, 5435 (1999)

    Article  ADS  Google Scholar 

  68. F. Becattini, L. Ferroni, Eur. Phys. J. C 35, 243 (2004)

    Article  ADS  Google Scholar 

  69. F. Becattini, L. Ferroni, Eur. Phys. J. C 38, 225 (2004)

    Article  ADS  Google Scholar 

  70. L.E. Reichl, A Modern Course in Statistical Physics (Wiley, New York, 1998)

    MATH  Google Scholar 

  71. L.D. Landau, E.M. Lifschitz, Statistical Physics (Fizmatlit, Moscow, 2001)

    Google Scholar 

  72. W. Greiner, L. Neise, H. Stöcker, Thermodynamics and Statistical Mechanics (Springer, New York, 1997)

    Google Scholar 

  73. A. Keränen, F. Becattini, Phys. Rev. C 65, 044901 (2002)

    Article  ADS  Google Scholar 

  74. Particle Data Group, Phys. Rev. D 66 (2002)

  75. G. Torrieri, S. Steinke, W. Broniowski, W. Florkowski, J. Letessier, J. Rafelski, Comput. Phys. Commun. 167, 229 (2005)

    Article  ADS  Google Scholar 

  76. S. Wheaton, J. Cleymans, arXiv:hep-ph/0407174

  77. I.S. Gradshteyn, I.M. Ryzhik, Table of Integrals, Series, and Products (Academic, San Diego, 2000)

    MATH  Google Scholar 

  78. M. Abramowitz, I.A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables (Dover, New York, 1965)

    Google Scholar 

  79. W. Feller, An Introduction to Probability Theory and Its Applications, vol. I (Wiley, New York, 1968)

    MATH  Google Scholar 

  80. W. Feller, An Introduction to Probability Theory and Its Applications, vol. II (Wiley, New York, 1970)

    Google Scholar 

  81. B. Hughes, Random Walks and Random Environments, vol. I (Clarendon, Oxford, 1995)

    MATH  Google Scholar 

  82. B. Hughes, Random Walks and Random Environments, vol. II (Clarendon, Oxford, 1996)

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Helmholtz Research School, University of Frankfurt, Frankfurt, Germany

    M. Hauer

  2. University of Cape Town, Cape Town, South Africa

    M. Hauer

  3. Bogolyubov Institute for Theoretical Physics, Kiev, Ukraine

    V. V. Begun & M. I. Gorenstein

  4. Frankfurt Institute for Advanced Studies, Frankfurt, Germany

    M. I. Gorenstein

Authors
  1. M. Hauer
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. V. V. Begun
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. M. I. Gorenstein
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to M. Hauer.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Hauer, M., Begun, V.V. & Gorenstein, M.I. Multiplicity distributions in canonical and microcanonical statistical ensembles. Eur. Phys. J. C 58, 83–110 (2008). https://doi.org/10.1140/epjc/s10052-008-0724-1

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1140/epjc/s10052-008-0724-1

PACS

Search

Navigation