- Physical Review Journals
All JournalsPhysics Magazine

Physical Review D

Reuse & Permissions

It is not necessary to obtain permission to reuse this article or its components as it is available under the terms of the Creative Commons Attribution 4.0 International license. This license permits unrestricted use, distribution, and reproduction in any medium, provided attribution to the author(s) and the published article's title, journal citation, and DOI are maintained. Please note that some figures may have been included with permission from other third parties. It is your responsibility to obtain the proper permission from the rights holder directly for these figures.

Export citation

Export citation

Choose format for download:

Download Citation
  • Open Access

Gaussian generally covariant hydrodynamics

G. M. Sampaio*, G. Rabelo-Soares†, and G. Torrieri‡

  • *Contact author: g224225@dac.unicamp.br
  • †Contact author: grsoares@ifi.unicamp.br
  • ‡Contact author: torrieri@ifi.unicamp.br
PDF

Phys. Rev. D 112, 056002 – Published 2 September, 2025

DOI: https://doi.org/10.1103/67x1-knzr

Abstract

We develop a version of fluctuating relativistic hydrodynamics in a way very different from the usual derivation: Instead of treating it as a coarse-grained deterministic theory expanded in gradients of equilibrium quantities, we treat it as a stochastic theory, characterized by partition functions in each cells, expanded in cumulants. We show that the Gaussian ansatz allows us, via the gravitational Ward identities acting as a constraint between the variance and the average, to maintain full general covariance, with hydrodynamic flow emerging as an approximate Killing vector. If the symmetry of ideal hydrodynamics, volume-preserving diffeomorphisms, is preserved, we show that linear response formulas are also generally covariant. We discuss our results and argue that in this approach, the applicability of the effective theory is parametrized around a very different quantity than the Knudsen number, offering hope of understanding the applicability of hydrodynamics to small systems.

View figure in article

Physics Subject Headings (PhySH)

Article Text

References (89)

  1. R. Derradi de Souza, T. Koide, and T. Kodama, Prog. Part. Nucl. Phys. 86, 35 (2016).
  2. S. Jeon and U. Heinz, Int. J. Mod. Phys. E 24, 1530010 (2015).
  3. M. M. Disconzi, Living Rev. Relativity 27, 6 (2024).
  4. G. S. Rocha, D. Wagner, G. S. Denicol, J. Noronha, and D. H. Rischke, Entropy 26, 189 (2024).
  5. D. Teaney, Phys. Rev. D 74, 045025 (2006).
  6. J. L. Nagle and W. A. Zajc, Annu. Rev. Nucl. Part. Sci. 68, 211 (2018).
  7. G. Torrieri, SciPost Phys. 16, 070 (2024).
  8. B. Betz, D. Henkel, and D. H. Rischke, Prog. Part. Nucl. Phys. 62, 556 (2009).
  9. F. Becattini, M. Buzzegoli, and A. Palermo, J. High Energy Phys. 02 (2021) 101.
  10. P. Kovtun, J. Phys. A 45, 473001 (2012).
  11. R. P. Geroch and L. Lindblom, Phys. Rev. D 41, 1855 (1990).
  12. A. Pandya, E. R. Most, and F. Pretorius, Phys. Rev. D 106, 123036 (2022).
  13. R. E. Hoult and P. Kovtun, Phys. Rev. D 106, 066023 (2022).
  14. J. Noronha, M. Spaliński, and E. Speranza, Phys. Rev. Lett. 128, 252302 (2022).
  15. W. Israel and J. M. Stewart, Ann. Phys. (N.Y.) 118, 341 (1979).
  16. A. Chiarini, J. Sammet, and M. Shokri, arXiv:2412.19854.
  17. L. D. Landau and E. M. Lifshitz, J. Exp. Theor. Phys. 32, 618 (1957).
  18. X. An, G. Basar, M. Stephanov, and H. U. Yee, Phys. Rev. C 108, 034910 (2023).
  19. C. Chattopadhyay, J. Ott, T. Schaefer, and V. V. Skokov, Phys. Rev. D 111, 034026 (2025).
  20. N. Mullins, M. Hippert, L. Gavassino, and J. Noronha, Phys. Rev. D 108, 116019 (2023).
  21. L. Gavassino, N. Mullins, and M. Hippert, Phys. Rev. D 109, 125002 (2024).
  22. H. Liu and P. Glorioso, Proc. Sci. TASI2017 (2018) 008 [arXiv:1805.09331].
  23. F. M. Haehl, R. Loganayagam, and M. Rangamani, J. High Energy Phys. 05 (2015) 060.
  24. A. Jain and P. Kovtun, Phys. Rev. Lett. 128, 071601 (2022).
  25. J. Bhambure, R. Singh, and D. Teaney, Phys. Rev. C 111, 064909 (2025).
  26. P. B. Arnold, Int. J. Mod. Phys. E 16, 2555 (2007).
  27. G. D. Moore, arXiv:2010.15704.
  28. L. Gavassino, Phys. Rev. Res. 6, L042043 (2024).
  29. G. S. Rocha, I. Danhoni, K. Ingles, G. S. Denicol, and J. Noronha, Phys. Rev. D 110, 076003 (2024).
  30. S. Grozdanov and J. Polonyi, Phys. Rev. D 91, 105031 (2015).
  31. D. Montenegro and G. Torrieri, Phys. Rev. D 94, 065042 (2016).
  32. G. Torrieri, J. High Energy Phys. 02 (2021) 175.
  33. G. Torrieri, Eur. Phys. J. A 56, 121 (2020).
  34. T. Dore, L. Gavassino, D. Montenegro, M. Shokri, and G. Torrieri, Ann. Phys. (N.Y.) 442, 168902 (2022).
  35. F. Taghinavaz and G. Torrieri, Phys. Rev. D 110, 056019 (2024).
  36. G. Torrieri, Phys. Rev. D 109, L051903 (2024).
  37. C. Barcelo, S. Liberati, and M. Visser, Living Rev. Relativity 8, 12 (2005).
  38. R. Jackiw, V. P. Nair, S. Y. Pi, and A. P. Polychronakos, J. Phys. A 37, R327 (2004).
  39. S. Dubovsky, T. Gregoire, A. Nicolis, and R. Rattazzi, J. High Energy Phys. 03 (2006) 025.
  40. D. T. Son (private communication).
  41. A. Kovner and J. G. Milhano, arXiv:hep-ph/0406165.
  42. G. Jona-Lasinio, Phys. Rep. 352, 439 (2001).
  43. D. Forster, Hydrodynamic Fluctuations, Broken Symmetry and Correlation Functions (Addison-Wesley, Reading, MA, 1990).
  44. L. P. Kadanoff and P. C. Martin, Ann. Phys. (N.Y.) 24, 419 (1963).
  45. M.Bonanca, Phys. Rev. E 78, 031107 (2008).
  46. F. Becattini, M. Buzzegoli, and E. Grossi, Particles 2, 197 (2019).
  47. J. Zinn-Justin, Int. Ser. Monogr. Phys. 113, 1 (2002).
  48. S. Deser and D. Boulware, J. Math. Phys. (N.Y.) 8, 1468 (1967).
  49. N. Mullins, M. Hippert, and J. Noronha, Phys. Rev. Lett. 134, 232302 (2025).
  50. L. Gavassino, M. Antonelli, and B. Haskell, Phys. Rev. Lett. 128, 010606 (2022).
  51. E. T. Jaynes, Probability Theory the Logic of Science (Cambridge University Press, Cambridge, England, 2003).
  52. T. S. Biro, G. Purcsel, and B. Muller, arXiv:hep-ph/0311301.
  53. E. T. Jaynes, Phys. Rev. 106, 620 (1957).
  54. F. Becattini, arXiv:0901.3643.
  55. M. Blake, H. Lee, and H. Liu, J. High Energy Phys. 10 (2018) 127.
  56. S. Grozdanov, P. K. Kovtun, A. O. Starinets, and P. Tadić, J. High Energy Phys. 11 (2019) 097.
  57. L. D. Landau and E. M. Lifshitz, Statistical Physics (Butterworth-Heinemann, 1980).
  58. K.Huang, Statistical Mechanics, 2nd ed. (Wiley, 1991).
  59. A. Y. Khinchin, Mathematical Foundations of Statistical Mechanics (Courier Corporation, 1949).
  60. A. Nicolis, R. Penco, F. Piazza, and R. Rattazzi, J. High Energy Phys. 06 (2015) 155.
  61. L. Gavassino, M. M. Disconzi, and J. Noronha, Phys. Rev. D 109, 096041 (2024).
  62. K. Jensen and A. Karch, J. High Energy Phys. 04 (2015) 155.
  63. Mohammad Farazmand and Mattia Serra, arXiv:1807.02726 and references therein.
  64. D. Bernard, K. Gawedzki, and A. Kupiainen, J. Stat. Phys. 90, 519 (1998).
  65. Dmytro Bandak, Alexei Mailybaev, Gregory L. Eyink, and Nigel Goldenfeld, arXiv:2401.13881.
  66. Alexei Mailybaev, Nonlinearity 35, 3630 (2022).
  67. Alex Hansen, Eirik G. Flekkøy, Santanu Sinha, and Per Arne Slotte, Adv. Water Resour. 171, 104336 (2023).
  68. Johanna Müller, S. Hermann, F. Sammüller, and M. Schmidt, Phys. Rev. Lett. 133, 217101 (2024).
  69. F. Becattini, J. Liao, and M. Lisa, Lect. Notes Phys. 987, 1 (2021).
  70. A. D. Gallegos, U. Gürsoy, and A. Yarom, SciPost Phys. 11, 041 (2021).
  71. A. Das, W. Florkowski, R. Ryblewski, and R. Singh, Phys. Rev. D 103, L091502 (2021).
  72. S. Jeon, EPJ Web Conf. 276, 01010 (2023).
  73. T. Brauner, Phys. Scr. 95, 035004 (2020).
  74. D. Montenegro and G. Torrieri, Phys. Rev. D 100, 056011 (2019).
  75. M. Visser and C. Barcelo, arXiv:gr-qc/0001099.
  76. M. L. W. Basso, J. Maziero, and L. C. Céleri, Phys. Rev. Lett. 134, 050406 (2025).
  77. J. H. Pei, J. F. Chen, and H. T. Quan, Phys. Rev. Lett. 134, 237102 (2025).
  78. R. P. Feynman, F. B. Morinigo, W. G. Wagner, and B. Hatfield, Feynman Lectures on Gravitation, 1st ed. (CRC Press, Boca Raton, 2003), 10.1201/9780429502859.
  79. E. Alvarez and E. Velasco-Aja, arXiv:2301.07641.
  80. T. Jacobson, Phys. Rev. Lett. 75, 1260 (1995).
  81. G. Torrieri, Eur. Phys. J. Special Topics 232, 3505 (2023).
  82. G. Torrieri, Int. J. Mod. Phys. D 34, 2550016 (2025).
  83. D. Wagner and L. Gavassino, Phys. Rev. D 109, 016019 (2024).
  84. W. Rietdijk, Fundam. Theor. Phys. 153, 101 (2007), and references therein
  85. S. Mukherjee, R. Venugopalan, and Y. Yin, Phys. Rev. Lett. 117, 222301 (2016).
  86. S. Popescu, A. J. Short, and A. Winter, Nat. Phys. 2, 754 (2006).
  87. A. Florio, D. Frenklakh, S. Grieninger, D. E. Kharzeev, A. Palermo, and S. Shi, arXiv:2506.14983.
  88. A. Peres and D. R. Terno, Rev. Mod. Phys. 76, 93 (2004).
  89. M. S. Guimaraes, I. Roditi, and S. P. Sorella, Universe 10, 396 (2024).

Outline

Information

Phys. Rev. D 112, 056002– Published 2 September, 2025
  • Received 7 May 2025
  • Accepted 25 July 2025
Crossmark - Check for updates button
Creative Commons Logo

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI. Funded by SCOAP3.

Published by the American Physical Society

Sign In to Your Journals Account

Filter

Filter

Article Lookup

Enter a citation