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Comparison of Boltzmann equations with quantum dynamics for scalar fields

Manfred Lindner1,* and Markus Michael Müller1,2,†

  • 1Physik-Department T30d, Technische Universität München, James-Franck-Straße, 85748 Garching, Germany
  • 2Max-Planck-Institut für Physik (Werner-Heisenberg-Institut), Föhringer Ring 6, 80805 München, Germany

  • *Electronic address: lindner@ph.tum.de
  • †Electronic address: Markus.Michael.Mueller@ph.tum.de
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Phys. Rev. D 73, 125002 – Published 5 June, 2006

DOI: https://doi.org/10.1103/PhysRevD.73.125002

Abstract

Boltzmann equations are often used to study the thermal evolution of particle reaction networks. Prominent examples are the computation of the baryon asymmetry of the universe and the evolution of the quark-gluon plasma after relativistic heavy ion collisions. However, Boltzmann equations are only a classical approximation of the quantum thermalization process which is described by the so-called Kadanoff-Baym equations. This raises the question how reliable Boltzmann equations are as approximations to the full Kadanoff-Baym equations. Therefore, we present in this paper a detailed comparison between the Kadanoff-Baym and Boltzmann equations in the framework of a scalar Φ4 quantum field theory in 3+1 space-time dimensions. The obtained numerical solutions reveal significant discrepancies in the results predicted by both types of equations. Apart from quantitative discrepancies, on a qualitative level the universality respected by the Kadanoff-Baym equations is severely restricted in the case of Boltzmann equations. Furthermore, the Kadanoff-Baym equations strongly separate the time scales between kinetic and chemical equilibration. This separation of time scales is absent for the Boltzmann equation.

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References (55)

  1. M. Fukugita and T. Yanagida, Phys. Lett. B 174, 45 (1986).
  2. W. Buchmüller, P. Di Bari, and M. Plümacher, Ann. Phys. (N.Y.) 315, 305 (2005).
  3. Wilfried Buchmüller and Stefan Fredenhagen, Phys. Lett. B 483, 217 (2000).
  4. A. D. Sakharov, JETP Lett. 5, 24 (1967); Pis’ma Zh. Eksp. Teor. Fiz. 5, 32 (1967).
  5. I. Arsene et al. (BRAHMS Collaboration), Nucl. Phys. A757, 1 (2005).
  6. B. B. Back et al. (PHOBOS Collaboration), Nucl. Phys. A757, 28 (2005).
  7. J. Adams et al. (STAR Collaboration), Nucl. Phys. A757, 102 (2005).
  8. K. Adcox et al. (PHENIX Collaboration), Nucl. Phys. A757, 184 (2005).
  9. J. Berges, S. Borsányi, and C. Wetterich, Phys. Rev. Lett. 93, 142002 (2004).
  10. A. H. Mueller and D. T. Son, Phys. Lett. B 582, 279 (2004).
  11. Sangyong Jeon, Phys. Rev. C 72, 014907 (2005).
  12. Gordon Baym and Leo P. Kadanoff, Quantum Statistical Mechanics (Benjamin, New York, 1962).
  13. P. Danielewicz, Ann. Phys. (N.Y.) 152, 239 (1984).
  14. Yu. B. Ivanov, J. Knoll, and D. N. Voskresensky, Nucl. Phys. A672, 313 (2000).
  15. J. Knoll, Yu. B. Ivanov, and D. N. Voskresensky, Ann. Phys. (N.Y.) 293, 126 (2001).
  16. Jean-Paul Blaizot and Edmond Iancu, Phys. Rep. 359, 355 (2002).
  17. Jürgen Berges, Nucl. Phys. A699, 847 (2002).
  18. Gert Aarts and Jürgen Berges, Phys. Rev. D 64, 105010 (2001).
  19. Edward W. Kolb and Stephen Wolfram, Nucl. Phys. B172, 224 (1980).
  20. Edward W. Kolb Michael S. Turnerand , The Early Universe (Addison-Wesley, Reading, MA, 1990).
  21. H. S. Köhler, Phys. Rev. C 51, 3232 (1995).
  22. K. Morawetz and H. S. Köhler, Eur. Phys. J. A 4, 291 (1999).
  23. S. Juchem, W. Cassing, and C. Greiner, Phys. Rev. D 69, 025006 (2004).
  24. Gordon Baym and Leo P. Kadanoff, Phys. Rev. 124, 287 (1961).
  25. Gordon Baym, Phys. Rev. 127, 1391 (1962).
  26. Yu. B. Ivanov, J. Knoll, and D. N. Voskresensky, Nucl. Phys. A657, 413 (1999).
  27. R. Jackiw, Phys. Rev. D 9, 1686 (1974).
  28. John M. Cornwall, R. Jackiw, and E. Tomboulis, Phys. Rev. D 10, 2428 (1974).
  29. E. Calzetta and B. L. Hu, Phys. Rev. D 37, 2878 (1988).
  30. Jürgen Berges and Jürgen Cox, Phys. Lett. B 517, 369 (2001).
  31. Jürgen Berges, Szabolcs Borsányi, and Julien Serreau, Nucl. Phys. B660, 51 (2003).
  32. Gert Aarts and Jürgen Berges, Phys. Rev. Lett. 88, 041603 (2002).
  33. Gert Aarts and Jose M. Martínez Resco, Phys. Rev. D 68, 085009 (2003).
  34. J. Berges, Sz. Borsányi, U. Reinosa, and J. Serreau, Phys. Rev. D 71, 105004 (2005).
  35. Jürgen Berges and Markus M. Müller, in Progress in Nonequilibrium Green’s Functions 2, edited by M. Bonitz and D. Semkat (World Scientific Publ., Singapore, 2003), p. 367.
  36. S. Juchem, W. Cassing, and C. Greiner, Nucl. Phys. A743, 92 (2004).
  37. Jürgen Berges and Szabolcs Borsányi, hep-th/0512010.
  38. Jurgen Berges and Szabolcs Borsanyi, hep-ph/0512155.
  39. Jürgen Berges, Szabolcs Borsányi, and Christof Wetterich, hep-ph/0505182.
  40. A. D. Dolgov, S. H. Hansen, and D. V. Semikoz, Nucl. Phys. B503, 426 (1997).
  41. Jürgen Berges, Introduction to Nonequilibrium Quantum Field Theory, AIP Conf. Proc. No. 739 (AIP, New York, 2005), p. 3.
  42. István Montvay and Gernot Münster, Quantum Fields on a Lattice (Cambridge University Press, Cambridge, England, 1994).
  43. Matteo Frigo and Steven G. Johnson, FFTW Reference Manual (Version 3.1, 2006)
  44. Alejandro Arrizabalaga, Jan Smit, and Anders Tranberg, Phys. Rev. D 72, 025014 (2005).
  45. Mark Galassi et al., GNU Scientific Library Reference Manual (Version 1.5, 2004). See also [55].
  46. Hendrik van Hees and Jörn Knoll, Phys. Rev. D 65, 025010 (2002).
  47. Hendrik van Hees and Jörn Knoll, Phys. Rev. D 65, 105005 (2002).
  48. Hendrik van Hees and Jörn Knoll, Phys. Rev. D 66, 025028 (2002).
  49. Jean-Paul Blaizot, Edmond Iancu, and Urko Reinosa, Nucl. Phys. A736, 149 (2004).
  50. Jürgen Berges, Szabolcs Borsányi, Urko Reinosa, and Julien Serreau, Ann. Phys. (N.Y.) 320, 344 (2005).
  51. Julian S. Schwinger, J. Math. Phys. (N.Y.) 2, 407 (1961).
  52. Pradip M. Bakshi and Kalyana T. Mahanthappa, J. Math. Phys. (N.Y.) 4, 1 (1963).
  53. Pradip M. Bakshi and Kalyana T. Mahanthappa, J. Math. Phys. (N.Y.) 4, 12 (1963).
  54. L. V. Keldysh, Sov. Phys. JETP 20, 1018 (1965); Zh. Eksp. Teor. Fiz. 47, 1515 (1964).
  55. William H. Press et al., Numerical Recipes in C (Cambridge University Press, Cambridge, England, 1988).

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Phys. Rev. D 73, 125002– Published 5 June, 2006
  • Received 13 January 2006

© 2006 American Physical Society

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