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Shear viscosity of a binary mixture for a relativistic fluid at high temperature

Gabriele Parisi*, Vincenzo Nugara†, Salvatore Plumari‡, and Vincenzo Greco§

  • *Contact author: gabriele.parisi@dfa.unict.it
  • †Contact author: vincenzo.nugara@phd.unict.it
  • ‡Contact author: salvatore.plumari@dfa.unict.it
  • §Contact author: greco@lns.infn.it
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Phys. Rev. D 113, 014001 – Published 2 January, 2026

DOI: https://doi.org/10.1103/qd2t-2sdp

Abstract

The determination of the shear viscosity is a central topic in various areas of modern physics. In particular, it is often necessary to evaluate the shear viscosity η of fluids made up of more than one species, all interacting with different cross sections. Since it may be difficult to extract information on the interaction among different species, various combinations of the viscosities of the individual components are often used. We work in the Chapman-Enskog framework and investigate binary mixtures, by comparing such single component combinations with a full 2-component formalism: we find that, in most cases, the full viscosity is well approximated by a weighted linear average of the single component viscosities, although this result is far from being general. Moreover, we validate our 2-component Chapman-Enskog results for η by comparing them with an independent numerical simulation of the Boltzmann equation, which estimates the shear viscosity via a Green-Kubo formula, in the case of a quasiparticle system that reproduces lattice QCD thermodynamics. We see that the temperature dependence of η/s of such a system of quarks and gluons is not well described by combinations of the individual components, highlighting the importance of interspecies scattering.

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References (77)

  1. L. Landau and E. Lifshitz, in Fluid Mechanics, 2nd ed. (Pergamon, Oxford, 1987).
  2. P. S. Shternin and D. G. Yakovlev, Phys. Rev. D 78, 063006 (2008).
  3. H. F. Zhang, U. Lombardo, and W. Zuo, Phys. Rev. C 82, 015805 (2010).
  4. D. Jaccarino, S. Plumari, V. Greco, U. Lombardo, and A. B. Santra, Phys. Rev. D 85, 103001 (2012).
  5. G. Bruun and H. Smith, Phys. Rev. A 72, 043605 (2005).
  6. G. Rupak and T. Schäfer, Phys. Rev. A 76, 053607 (2007).
  7. T. Enss, R. Haussmann, and W. Zwerger, Ann. Phys. (Amsterdam) 326, 770 (2011).
  8. A. Levchenko and J. Schmalian, Ann. Phys. (Amsterdam) 419, 168218 (2020).
  9. C. Manuel and L. Tolos, Phys. Rev. D 84, 123007 (2011).
  10. C. Manuel and L. Tolos, Phys. Rev. D 88, 043001 (2013).
  11. M. Mannarelli, C. Manuel, and L. Tolos, Ann. Phys. (Amsterdam) 336, 12 (2013).
  12. M. Mannarelli, C. Manuel, and L. Tolos, arXiv:1201.4006.
  13. J. Noronha-Hostler, J. Noronha, and C. Greiner, Phys. Rev. C 86, 024913 (2012).
  14. A. El, F. Lauciello, C. Wesp, Z. Xu, and C. Greiner, Nucl. Phys. A925, 150 (2014).
  15. A. Zaccone, Nucl. Phys. B1000, 116483 (2024).
  16. A. Zaccone, Phys. Rev. E 110, L052101 (2024).
  17. P. Kovtun, D. T. Son, and A. O. Starinets, Phys. Rev. Lett. 94, 111601 (2005).
  18. T. Haxhimali, R. E. Rudd, W. H. Cabot, and F. R. Graziani, Phys. Rev. E 92, 053110 (2015).
  19. S. Trabucco, L. Lepori, M. Luisa Chiofalo, and M. Mannarelli, Phys. Rev. D 111, 125024 (2025).
  20. P. Arnault, High Energy Density Phys. 9, 711 (2013).
  21. S. R. De Groot, W. A. Van Leeuwen, and C. G. Van Weert, Relativistic Kinetic Theory. Principles and Applications (North-Holland Publishing Company, Amsterdam, 1980).
  22. L. Jehring, S. Chapman, and T. G. Cowling, Z. Angew. Math. Mech. 72, 610 (1992).
  23. W. Israel, J. Math. Phys. (N.Y.) 4, 1163 (1963).
  24. A. Wiranata and M. Prakash, Phys. Rev. C 85, 054908 (2012).
  25. M. Prakash, M. Prakash, R. Venugopalan, and G. Welke, Phys. Rep. 227, 321 (1993).
  26. G. S. Denicol, E. Molnár, H. Niemi, and D. H. Rischke, Eur. Phys. J. A 48, 170 (2012).
  27. A. Wiranata, V. Koch, M. Prakash, and X. N. Wang, Phys. Rev. C 88, 044917 (2013).
  28. X.-G. Deng, D.-Q. Fang, and Y.-G. Ma, Prog. Part. Nucl. Phys. 136, 104095 (2024).
  29. S. Sarkar and R. Sharma, Phys. Rev. D 96, 094025 (2017).
  30. F. Frascà, A. Beraudo, and M. Strickland, Nucl. Phys. A1055, 123008 (2025).
  31. M. I. Gorenstein, M. Hauer, and O. N. Moroz, Phys. Rev. C 77, 024911 (2008).
  32. M. S. Green, J. Chem. Phys. 22, 398 (1954).
  33. R. Kubo, M. Yokota, and S. Nakajima, J. Phys. Soc. Jpn. 12, 1203 (1957).
  34. D. Zubarev, V. Morozov, and G. Röpke, Relaxation and Hydrodynamic Processes, Statistical Mechanics of Nonequilibrium Processes (Akademie Verlag, Berlin, 1996).
  35. S. Plumari, A. Puglisi, F. Scardina, and V. Greco, Phys. Rev. C 86, 054902 (2012).
  36. G. Ferini, M. Colonna, M. Di Toro, and V. Greco, Phys. Lett. B 670, 325 (2009).
  37. F. Scardina, M. Colonna, S. Plumari, and V. Greco, Phys. Lett. B 724, 296 (2013).
  38. A. Puglisi, S. Plumari, and V. Greco, Phys. Rev. D 90, 114009 (2014).
  39. F. Scardina, D. Perricone, S. Plumari, M. Ruggieri, and V. Greco, Phys. Rev. C 90, 054904 (2014).
  40. S. Plumari, G. L. Guardo, V. Greco, and J.-Y. Ollitrault, Nucl. Phys. A941, 87 (2015).
  41. S. Plumari, G. L. Guardo, F. Scardina, and V. Greco, Phys. Rev. C 92, 054902 (2015).
  42. F. Scardina, S. K. Das, V. Minissale, S. Plumari, and V. Greco, Phys. Rev. C 96, 044905 (2017).
  43. S. Plumari, Eur. Phys. J. C 79, 2 (2019).
  44. S. Plumari, G. Coci, V. Minissale, S. K. Das, Y. Sun, and V. Greco, Phys. Lett. B 805, 135460 (2020).
  45. Y. Sun, S. Plumari, and V. Greco, Eur. Phys. J. C 80, 16 (2020).
  46. M. L. Sambataro, S. Plumari, and V. Greco, Eur. Phys. J. C 80, 1140 (2020).
  47. M. L. Sambataro, Y. Sun, V. Minissale, S. Plumari, and V. Greco, Eur. Phys. J. C 82, 833 (2022).
  48. M. L. Sambataro, V. Minissale, S. Plumari, and V. Greco, Phys. Lett. B 849, 138480 (2024).
  49. A. Gabbana, S. Plumari, G. Galesi, V. Greco, D. Simeoni, S. Succi, and R. Tripiccione, Phys. Rev. C 101, 064904 (2020).
  50. L. Oliva, S. Plumari, and V. Greco, J. High Energy Phys. 05 (2021) 034.
  51. V. Nugara, S. Plumari, L. Oliva, and V. Greco, Eur. Phys. J. C 84, 861 (2024).
  52. V. Nugara, V. Greco, and S. Plumari, Eur. Phys. J. C 85, 311 (2025).
  53. V. Nugara, N. Borghini, V. Greco, and S. Plumari, arXiv:2509.05495.
  54. C.-Y. Wong, Phys. Rev. C 25, 1460 (1982).
  55. H. B. Meyer, Phys. Rev. D 76, 101701 (2007).
  56. H. B. Meyer, Nucl. Phys. A830, 641C (2009).
  57. S. W. Mages, S. Borsányi, Z. Fodor, A. Schäfer, and K. Szabó, Proc. Sci. LATTICE2014 (2015) 232.
  58. S. Borsányi, Z. Fodor, M. Giordano, S. D. Katz, A. Pasztor, C. Ratti, A. Schäfer, K. K. Szabo, and B. C. Tóth, Phys. Rev. D 98, 014512 (2018).
  59. L. Altenkort, A. M. Eller, A. Francis, O. Kaczmarek, L. Mazur, G. D. Moore, and H.-T. Shu, Phys. Rev. D 108, 014503 (2023).
  60. J. Noronha-Hostler, L. Yan, F. G. Gardim, and J.-Y. Ollitrault, Phys. Rev. C 93, 014909 (2016).
  61. H. Roch and N. Borghini, Eur. Phys. J. C 81, 380 (2021).
  62. V. Ozvenchuk, O. Linnyk, M. I. Gorenstein, E. L. Bratkovskaya, and W. Cassing, Phys. Rev. C 87, 064903 (2013).
  63. S. Typel, G. Ropke, T. Klahn, D. Blaschke, and H. H. Wolter, Phys. Rev. C 81, 015803 (2010).
  64. R. Wang, Y.-G. Ma, L.-W. Chen, C. M. Ko, K.-J. Sun, and Z. Zhang, Phys. Rev. C 108, L031601 (2023).
  65. G. Parisi, Shear viscosity of a binary mixture for a relativistic fluid at high temperature [Data set] (2025), 10.5281/zenodo.17531902.
  66. S. Plumari, V. Minissale, S. K. Das, G. Coci, and V. Greco, Eur. Phys. J. C 78, 348 (2018).
  67. S. Plumari, W. M. Alberico, V. Greco, and C. Ratti, Phys. Rev. D 84, 094004 (2011).
  68. S. Borsanyi, G. Endrodi, Z. Fodor, A. Jakovac, S. D. Katz, S. Krieg, C. Ratti, and K. K. Szabo, J. High Energy Phys. 11 (2010) 077.
  69. O. Kaczmarek and F. Zantow, Phys. Rev. D 71, 114510 (2005).
  70. M. L. Sambataro, V. Greco, G. Parisi, and S. Plumari, Eur. Phys. J. C 84, 881 (2024).
  71. P. Moreau, O. Soloveva, L. Oliva, T. Song, W. Cassing, and E. Bratkovskaya, Phys. Rev. C 100, 014911 (2019).
  72. M. Gell-Mann, Phys. Rev. 125, 1067 (1962).
  73. A. J. MacFarlane, A. Sudbery, and P. H. Weisz, Commun. Math. Phys. 11, 77 (1968).
  74. R. Mertig, M. Bohm, and A. Denner, Comput. Phys. Commun. 64, 345 (1991).
  75. V. Shtabovenko, R. Mertig, and F. Orellana, Comput. Phys. Commun. 207, 432 (2016).
  76. V. Shtabovenko, R. Mertig, and F. Orellana, Comput. Phys. Commun. 256, 107478 (2020).
  77. R. K. Ellis, W. J. Stirling, and B. R. Webber, QCD and Collider Physics (Cambridge University Press, Cambridge, England, 2011), Vol. 8.

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Phys. Rev. D 113, 014001– Published 2 January, 2026
  • Received 5 November 2025
  • Accepted 10 December 2025
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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.

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