- Physical Review Journals
All JournalsPhysics Magazine
Access Provided by Xinjiang University

Physical Review D

Export citation

Export citation

Choose format for download:

Download Citation
  • Editors' Suggestion
  • Access by Xinjiang University

Divergence and resummation of the moment expansion for an ultrarelativistic gas in Bjorken flow

Caio V. P. de Brito1,2,3,4,*, David Wagner5,†, Gabriel S. Denicol1,‡, and Dirk H. Rischke2,6,§

  • 1Instituto de Física, Universidade Federal Fluminense, Avenida General Milton Tavares de Souza, S/N, 24210-346, Gragoatá, Niterói, Rio de Janeiro, Brazil
  • 2Institute for Theoretical Physics, Goethe University, Max-von-Laue-Str. 1, D-60438 Frankfurt am Main, Germany
  • 3University of Jyväskylä, Department of Physics, P.O.B. 35, FI-40014 University of Jyväskylä, Finland
  • 4Helsinki Institute of Physics, P.O.B. 64, FI-00014 University of Helsinki, Finland
  • 5Università degli studi di Firenze, Via Giovanni Sansone 1, I-50019 Sesto Fiorentino (Florence), Italy
  • 6Helmholtz Research Academy Hesse for FAIR, Campus Riedberg, Max-von-Laue-Strasse 12, D-60438 Frankfurt am Main, Germany

  • *Contact author: caio.vp.debrito@jyu.fi
  • †Contact author: david.wagner@unifi.it
  • ‡Contact author: gsdenicol@id.uff.br
  • §Contact author: drischke@itp.uni-frankfurt.de
PDF

Phys. Rev. D 113, 096007 – Published 8 May, 2026

DOI: https://doi.org/10.1103/29mk-wwkg

Abstract

In this work, we demonstrate for the first time that the moment expansion for an ultrarelativistic gas undergoing Bjorken flow diverges. We then show how this series can be resummed using the Borel-Padé method and use this to determine the single-particle distribution function of the gas. Finally, we compare the exact resummed solution of the single-particle distribution function with solutions of the Boltzmann equation in the hydrodynamic limit and verify that the system displays considerable deviations from local equilibrium.

View figure in article

Physics Subject Headings (PhySH)

Article Text

References (56)

  1. C. Gale, S. Jeon, and B. Schenke, Int. J. Mod. Phys. A 28, 1340011 (2013).
  2. U. Heinz and R. Snellings, Annu. Rev. Nucl. Part. Sci. 63, 123 (2013).
  3. C. Shen and L. Yan, Nucl. Sci. Tech. 31, 122 (2020).
  4. J.-F. Paquet, J. Phys. G 51, 103001 (2024).
  5. M. Chandra, C. F. Gammie, F. Foucart, and E. Quataert, Astrophys. J. 810, 162 (2015).
  6. E. R. Most, J. Noronha, and A. A. Philippov, Mon. Not. R. Astron. Soc. 514, 4989 (2022).
  7. G. S. Denicol, H. Niemi, E. Molnár, and D. H. Rischke, Phys. Rev. D 85, 114047 (2012); 91, 039902(E) (2015).
  8. G. S. Denicol, E. Molnár, H. Niemi, and D. H. Rischke, Eur. Phys. J. A 48, 170 (2012).
  9. W. Florkowski, R. Ryblewski, and M. Strickland, Phys. Rev. C 88, 024903 (2013).
  10. G. S. Denicol, W. Florkowski, R. Ryblewski, and M. Strickland, Phys. Rev. C 90, 044905 (2014).
  11. G. S. Denicol, U. W. Heinz, M. Martinez, J. Noronha, and M. Strickland, Phys. Rev. Lett. 113, 202301 (2014).
  12. M. Strickland, J. Noronha, and G. Denicol, Phys. Rev. D 97, 036020 (2018).
  13. W. Florkowski, M. P. Heller, and M. Spalinski, Rep. Prog. Phys. 81, 046001 (2018).
  14. M. P. Heller, A. Serantes, M. Spaliński, V. Svensson, and B. Withers, SciPost Phys. 10, 123 (2021).
  15. S. Bhadury, W. Florkowski, A. Jaiswal, A. Kumar, and R. Ryblewski, Phys. Rev. Lett. 129, 192301 (2022).
  16. D. Wagner, A. Palermo, and V. E. Ambruş, Phys. Rev. D 106, 016013 (2022).
  17. J. A. Fotakis, E. Molnár, H. Niemi, C. Greiner, and D. H. Rischke, Phys. Rev. D 106, 036009 (2022).
  18. C. V. P. de Brito and G. S. Denicol, Phys. Rev. D 108, 096020 (2023).
  19. G. S. Rocha, C. V. P. de Brito, and G. S. Denicol, Phys. Rev. D 108, 036017 (2023).
  20. C. Chattopadhyay, U. Heinz, and T. Schaefer, Phys. Rev. C 108, 034907 (2023).
  21. D. Dash, S. Jaiswal, S. Bhadury, and A. Jaiswal, Phys. Rev. C 108, 064913 (2023).
  22. K. Kushwah and G. S. Denicol, Phys. Rev. D 109, 096021 (2024).
  23. F. Frascà, A. Beraudo, and M. Strickland, Nucl. Phys. A1055, 123008 (2025).
  24. M. Strickland, Int. J. Mod. Phys. E 33, 2430004 (2024).
  25. S. Chapman and T. G. Cowling, The Mathematical Theory of Non-Uniform Gases (Cambridge University Press, Cambridge, England, 1970).
  26. C. Cercignani and G. M. Kremer, The Relativistic Boltzmann Equation: Theory and Applications (Springer, New York, 2002).
  27. H. Grad, Commun. Pure Appl. Math. 2, 331 (1949).
  28. W. Israel and J. M. Stewart, Ann. Phys. (N.Y.) 118, 341 (1979).
  29. G. S. Denicol and D. H. Rischke, Microscopic Foundations of Relativistic Fluid Dynamics (Springer, New York, 2021).
  30. W. A. Hiscock and L. Lindblom, Phys. Rev. D 31, 725 (1985).
  31. G. S. Denicol, T. Kodama, T. Koide, and P. Mota, J. Phys. G 35, 115102 (2008).
  32. S. Pu, T. Koide, and D. H. Rischke, Phys. Rev. D 81, 114039 (2010).
  33. C. V. Brito and G. S. Denicol, Phys. Rev. D 102, 116009 (2020).
  34. C. V. P. de Brito and G. S. Denicol, Phys. Rev. D 110, 036017 (2024).
  35. J.-F. Paquet, C. Shen, G. S. Denicol, M. Luzum, B. Schenke, S. Jeon, and C. Gale, Phys. Rev. C 93, 044906 (2016).
  36. G. Vujanovic, J.-F. Paquet, G. S. Denicol, M. Luzum, S. Jeon, and C. Gale, Phys. Rev. C 94, 014904 (2016).
  37. D. Molnar and Z. Wolff, Phys. Rev. C 95, 024903 (2017).
  38. D. Everett et al. (JETSCAPE Collaboration), Phys. Rev. C 103, 054904 (2021).
  39. D. Bazow, G. S. Denicol, U. Heinz, M. Martinez, and J. Noronha, Phys. Rev. Lett. 116, 022301 (2016).
  40. J. D. Bjorken, Phys. Rev. D 27, 140 (1983).
  41. É. Borel, Ann. Sci. Ec. Norm. Super. 16, 9 (1899).
  42. H. Padé, Ann. Sci. Ec. Norm. Super. 9, 3 (1892).
  43. G. Baym, B. L. Friman, J. P. Blaizot, M. Soyeur, and W. Czyz, Nucl. Phys. A407, 541 (1983).
  44. M. P. Heller and M. Spalinski, Phys. Rev. Lett. 115, 072501 (2015).
  45. U. Heinz, D. Bazow, G. S. Denicol, M. Martinez, M. Nopoush, J. Noronha, R. Ryblewski, and M. Strickland, Nucl. Part. Phys. Proc. 276–278, 193 (2016).
  46. S. Chen and S. Shi, Phys. Rev. C 109, L051901 (2024).
  47. V. Nugara, S. Plumari, L. Oliva, and V. Greco, Eur. Phys. J. C 84, 861 (2024).
  48. S. Jaiswal, C. Chattopadhyay, L. Du, U. Heinz, and S. Pal, Phys. Rev. C 105, 024911 (2022).
  49. A. Dash, M. Shokri, L. Rezzolla, and D. H. Rischke, Phys. Rev. D 107, 056003 (2023).
  50. J. Anderson and H. Witting, Physica (Utrecht) 74, 466 (1974).
  51. G. S. Rocha, G. S. Denicol, and J. Noronha, Phys. Rev. Lett. 127, 042301 (2021).
  52. L. Landau and E. Lifshitz, Fluid Mechanics, Course of Theoretical Physics, Vol. 6, (Addison-Wesley, Reading, MA, 1959).
  53. G. A. Baker Jr. and P. Graves-Morris, Padé Approximants, Encyclopedia of Mathematics and its Applications, (Cambridge University Press, Cambridge, England, 1996), 2nd ed., Vol. 59.
  54. C. V. P. de Brito, D. Wagner, G. S. Denicol, and D. H. Rischke, https://github.com/caiovpdebrito/resummation-moment-expansion.
  55. M. Tatari and M. Hamadi, arXiv:2011.02295.
  56. DLMF, NIST Digital Library of Mathematical Functions, https://dlmf.nist.gov/, Release 1.2.0 of 2024-03-15, f. W. J. Olver, A. B. Olde Daalhuis, D. W. Lozier, B. I. Schneider, R. F. Boisvert, C. W. Clark, B. R. Miller, B. V. Saunders, H. S. Cohl, and M. A. McClain, eds.

Outline

Information

Phys. Rev. D 113, 096007– Published 8 May, 2026
  • Received 6 November 2025
  • Accepted 19 February 2026
Crossmark - Check for updates button

© 2026 American Physical Society

Sign In to Your Journals Account

Filter

Filter

Article Lookup

Enter a citation