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QCD phase transition at finite temperature and chemical potential with nonextensive statistics

Zhi-Ying Qin1,2, Jia-Hao Shi1,3, Jin-Peng Zhang1, Jian Cao1, Bo Feng1, Wen-Chao Zhang1,*, Hua Zheng1, and Shi-Jun Mao4

  • 1School of Physics and Information Technology, Shaanxi Normal University, Xi’an 710119, China
  • 2School of Physics Science and Engineering, Tongji University, Shanghai 200092, China
  • 3Key Laboratory of Quark and Lepton Physics (MOE) and Institute of Particle Physics, Central China Normal University, Wuhan 430079, China
  • 4School of Physics, Xi’an Jiaotong University, Xi’an, Shaanxi 710049, China

  • *Contact author: wenchao.zhang@snnu.edu.cn
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Phys. Rev. D 112, 096022 – Published 19 November, 2025

DOI: https://doi.org/10.1103/4gzt-46t5

Abstract

The intrinsic fluctuations, memory effects, and long-range color interactions in high-energy nuclear collisions imply the presence of non-Markovian processes in the fireball evolution, which affects the thermalization process toward equilibrium and produces a nonextensive behavior. In order to investigate the nonequilibrium effect on the quantum chromodynamics (QCD) phase transition at finite temperature (T) and chemical potential (μ), we apply a nonextensive correction to the equation of state in the parton (hadron resonance) gas at high (low) temperature and interpolate these two equations of state with a smooth crossover. The nonextensive statistics is characterized by a nonextensivity parameter q, which measures the degrees of deviation from the thermal equilibrium. It is found that the dimensionless thermodynamic quantities such as the entropy density, the pressure, the energy density, the specific heat at constant volume, and the trace anomaly are sensitive to the deviation of q from unity, and they become large in both the hadronic and quark-gluon plasma phases with the increase of q. Moreover, this deviation leads to nontrivial corrections of the squared speed of sound [(cs2)q] in the vicinity of the critical point (Tc) and at lower temperatures. Additionally, these thermodynamic quantities are sensitive to the deviation of μ from zero. With increasing μ, they become enhanced in both phases. Specifically, for (cs2)q, the value increases near Tc but decreases at lower temperatures. Finally, we observe that our results with q=1 agree well with those from lattice QCD, the hadron resonance gas model, and the thermal-fist fit to the hadron yields in high-energy nuclear collisions in the low-temperature region up to T∼150  MeV.

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References (93)

  1. A. Chodos, R. L. Jaffe, K. Johnson, C. B. Thorn, and V. F. Weisskopf, New extended model of hadrons, Phys. Rev. D 9, 3471 (1974).
  2. Y. Nambu and G. Jona-Lasinio, Dynamical model of elementary particles based on an analogy with superconductivity. I, Phys. Rev. 122, 345 (1961).
  3. Y. Nambu and G. Jona-Lasinio, Dynamical model of elementary particles based on an analogy with superconductivity. II, Phys. Rev. 124, 246 (1961).
  4. M. N. Chernodub and A. S. Nedelin, Phase diagram of chirally imbalanced QCD matter, Phys. Rev. D 83, 105008 (2011).
  5. B. J. Schaefer, J. M. Pawlowski, and J. Wambach, Phase structure of the Polyakov-quark-meson model, Phys. Rev. D 76, 074023 (2007).
  6. B. J. Schaefer, M. Wagner, and J. Wambach, Thermodynamics of (2+1)-flavor QCD: Confronting models with lattice studies, Phys. Rev. D 81, 074013 (2010).
  7. C. D. Roberts and A. G. Williams, Dyson-Schwinger equations and their application to hadronic physics, Prog. Part. Nucl. Phys. 33, 477 (1994).
  8. C. D. Roberts and S. M. Schmidt, Dyson-Schwinger equations: Density, temperature and continuum strong QCD, Prog. Part. Nucl. Phys. 45, S1 (2000).
  9. C. Wetterich, Exact evolution equation for the effective potential, Phys. Lett. B 301, 90 (1993).
  10. N. Dupuis, L. Canet, A. Eichhorn, W. Metzner, J. M. Pawlowski, M. Tissier, and N. Wschebor, The nonperturbative functional renormalization group and its applications, Phys. Rep. 910, 1 (2021).
  11. J.-X. Shao, W.-J. Fu, and Y.-X. Liu, Chemical freeze-out parameters via a functional renormalization group approach, Phys. Rev. D 109, 034019 (2024).
  12. M. Asakawa and T. Hatsuda, What thermodynamics tells us about the QCD plasma, Phys. Rev. D 55, 4488 (1997).
  13. F. Karsch, K. Redlich, and A. Tawfik, Hadron resonance mass spectrum and lattice QCD thermodynamics, Eur. Phys. J. C 29, 549 (2003).
  14. P. Huovinen and P. Petreczky, QCD equation of state and hadron resonance gas, Nucl. Phys. A837, 26 (2010).
  15. A. Bazavov et al. (HotQCD Collaboration), Fluctuations and correlations of net baryon number, electric charge, and strangeness: A comparison of lattice QCD results with the hadron resonance gas model, Phys. Rev. D 86, 034509 (2012).
  16. A. Bazavov et al. (HotQCD Collaboration), Equation of state in (2+1)-flavor QCD, Phys. Rev. D 90, 094503 (2014).
  17. S. Borsanyi, Z. Fodor, C. Hoelbling, S. D. Katz, S. Krieg, and K. K. Szabó, Full result for the QCD equation of state with 2+1 flavors, Phys. Lett. B 730, 99 (2014).
  18. G. Wilk and Z. Wlodarczyk, Interpretation of the nonextensivity parameter q in some applications of Tsallis statistics and lévy distributions, Phys. Rev. Lett. 84, 2770 (2000).
  19. T. S. Biro and A. Jakovac, Power-law tails from multiplicative noise, Phys. Rev. Lett. 94, 132302 (2005).
  20. T. Osada and G. Wilk, Nonextensive hydrodynamics for relativistic heavy-ion collisions, Phys. Rev. C 77, 044903 (2008).
  21. M. Ruggeieri, Pooja, J. Prakash, and S. K. Das, Memory effects on energy loss and diffusion of heavy quarks in the quark-gluon plasma,, Phys. Rev. D 106, 034032 (2022).
  22. A. Dragoa, A. Lavagno, and P. Quarati, Nonextensive statistical effects on the relativistic nuclear equation of state, Physica (Amsterdam) 344A, 472 (2004).
  23. T. S. Biro and G. Purcsel, Nonextensive statistical effects on the relativistic nuclear equation of state, Phys. Rev. Lett. 95, 162302 (2005).
  24. T. S. Biro and G. Purcsel, Equilibration of two power-law tailed distributions in a parton cascade model, Phys. Lett. A 372, 1174 (2008).
  25. C. Tsallis, Possible generalization of Boltzmann-Gibbs statistics, J. Stat. Phys. 52, 479 (1988).
  26. A. Deppman, Thermodynamics with fractal structure, Tsallis statistics, and hadrons, Phys. Rev. D 93, 054001 (2016).
  27. A. Deppman, E. Megias, and D. P. Menezes, Fractals, nonextensive statistics, and QCD, Phys. Rev. D 101, 034109 (2020).
  28. J. Cleymans and D. Worku, Relativistic thermodynamics: Transverse momentum distributions in high-energy physics, Eur. Phys. J. A 48, 160 (2012).
  29. J. Cleymans and D. Worku, The Tsallis distribution in proton–-proton collisions at s=0.9 TeV at the LHC, J. Phys. G 39, 025006 (2012).
  30. J. Adams et al. (STAR Collaboration), Identified hadron spectra at large transverse momentum in pp and d+Au collisions at sNN=200  GeV, Phys. Lett. B 637, 161 (2006).
  31. A. Adare et al. (PHENIX Collaboration), Nuclear modification factors of ϕ mesons in d+Au, Cu+Cu, and Au+Au collisions at sNN=200 GeV, Phys. Rev. C 83, 024909 (2011).
  32. W.-C. Zhang and C.-B. Yang, Scaling behaviour of charged hadron distributions in pp and pp¯-collisions, J. Phys. G 41, 105006 (2014).
  33. S. Chatrchyan et al. (ALICE Collaboration), Study of the production of charged pions, kaons, and protons in pPb collisions at sNN=5.02  TeV, Eur. Phys. J. C 74, 2847 (2014).
  34. M. D. Azmi and J. Cleymans, Transverse momentum distributions at the LHC and Tsallis thermodynamics, Acta Phys. Pol. B Proc. Suppl. 7, 9 (2014).
  35. H. Zheng and L. Zhu, Transverse momentum distributions at the lhc and tsallis thermodynamicscan tsallis distribution fit all the particle spectra produced at RHIC and LHC?, Adv. High Energy Phys. 2015, 180491 (2015).
  36. W.-C. Zhang, Scaling behaviours of the pT spectra for identified hadrons in pp collisions, J. Phys. G 43, 015003 (2016).
  37. J. Cleymans, G. I. Lykasov, A. S. Parvan, A. S. Sorin, O. V. Teryaev, and D. Worku, Systematic properties of the Tsallis distribution: Energy dependence of parameters in high energy p−p collisions, Phys. Lett. B 723, 351 (2013).
  38. M. D. Azmi and J. Cleymans, Transverse momentum distributions in proton–proton collisions at LHC energies and Tsallis thermodynamics, J. Phys. G 41, 065001 (2014).
  39. M. D. Azmi and J. Cleymans, The Tsallis distribution at large transverse momenta, Eur. Phys. J. C 75, 430 (2015).
  40. L. Marques, J. Cleymans, and A. Deppman, Description of high-energy pp collisions using Tsallis thermodynamics: Transverse momentum and rapidity distributions,, Phys. Rev. D 91, 054025 (2015).
  41. H. Zheng, L. Zhu, and A. Bonasera, Systematic analysis of hadron spectra in p+p collisions using Tsallis distributions, Phys. Rev. D 92, 074009 (2015).
  42. A. Khuntia, S. Tripathy, R. Sahoo, and J. Cleymans, Multiplicity dependence of non-extensive parameters for strange and multi-strange particles in proton-proton collisions at s=7  TeV at the LHC, Eur. Phys. J. A 53, 103 (2017).
  43. L.-Y. Qiao, G.-R. Che, J.-B. Gu, H. Zheng, and W.-C. Zhang, Nuclear modification factor in Pb–Pb and p-Pb collisions using Boltzmann transport equation, J. Phys. G 47, 075101 (2020).
  44. G.-R. Che, J.-B. Gu, W.-C. Zhang, and H. Zheng, Identified particle spectra in Pb–Pb, Xe–Xe and p–Pb collisions with the Tsallis blast-wave model, J. Phys. G 48, 095103 (2021).
  45. J.-B. Gu, C.-Y. Li, Q. Wang, W.-C. Zhang, and H. Zheng, Collective expansion in pp collisions using the Tsallis statistics, J. Phys. G 49, 115101 (2022).
  46. F. S. Navarra, O. V. Utyuzh, G. Wilk, and Z. Wlodarczyk, Estimating inelasticity with the information theory approach, Phys. Rev. D 67, 114002 (2003).
  47. M. Alqahtani, N. Demir, and M. Strickland, Nonextensive hydrodynamics of boost-invariant plasmas, Eur. Phys. J. C 82, 973 (2022).
  48. K. Kyan and A. Monnai, QCD equation of state with Tsallis statistics for heavy-ion collisions, Phys. Rev. D 106, 054004 (2022).
  49. J.-H. Shi, Z.-Y. Qin, J.-P. Zhang, J. Cao, Z.-F. Jiang, W.-C. Zhang, and H. Zheng, Nonextensive (3+1)-dimensional hydrodynamics for relativistic heavy-ion collisions, Phys. Rev. D 111, 036010 (2025).
  50. P. Arnold, J. Lenaghan, G. D. Moore, and L. G. Yaffe, Apparent thermalization due to plasma instabilities in the quark-gluon plasma, Phys. Rev. Lett. 94, 072302 (2005).
  51. A. Takacs and D. Molnar, Suppression of elliptic flow without viscosity, arXiv:1906.12311.
  52. A. M. Teweldeberhan, H. G. Miller, and R. Tegen, Generalized statistics and the formation of a quark-gluon plasma, Int. J. Mod. Phys. E 12, 395 (2003).
  53. F. I. M. Pereira, R. Silva, and J. S. Alcaniz, Nonextensive effects on the relativistic nuclear equation of state, Phys. Rev. C 76, 015201 (2007).
  54. A. Lavagno, D. Pigato, and P. Quarati, Nonextensive statistical effects in the hadron to quark–gluon phase transition, J. Phys. G 37, 115102 (2010).
  55. J. Rozynek and G. Wilk, Nonextensive effects in the Nambu–Jona-Lasinio model of QCD, J. Phys. G 36, 125108 (2009).
  56. J. Rozynek and G. Wilk, Nonextensive Nambu-Jona-Lasinio Model of QCD matter, Eur. Phys. J. A 52, 13 (2016).
  57. Y.-P. Zhao, Thermodynamic properties and transport coefficients of QCD matter within the nonextensive Polyakov–Nambu–Jona-Lasinio model, Phys. Rev. D 101, 096006 (2020).
  58. Y.-P. Zhao, S.-Y. Zuo, and C.-M. Li, QCD chiral phase transition and critical exponents within the nonextensive Polyakov-Nambu-Jona-Lasinio model, Chin. Phys. C 45, 073105 (2021).
  59. R. Hagedorn, Statistical dynamics of strong interactions at high energies, Nuovo Cimento Suppl. 3, 147 (1965), http://cds.cern.ch/record/346206.
  60. J. Kapusta and C. Gale, Finite Temperature Field Theory: Principles and Applications, Cambridge Monographs on Mathematical Physics (Cambridge University Press, Cambridge, England, 2006).
  61. V. Vovchenko and H. Stoecker, Thermal-FIST: A package for heavy-ion collisions and hadronic equation of state, Comput. Phys. Commun. 244, 295 (2019).
  62. C. Tsallis, Possible generalization of Boltzmann-Gibbs statistics, J. Stat. Phys. 52, 479 (1988).
  63. S. Mitra, Thermodynamics and relativistic kinetic theory for q-generalized Bose–-Einstein and Fermi–-Dirac systems, Eur. Phys. J. C 78, 66 (2018).
  64. L. D. Landau and E. M. Lifshitz, Statistical Physics, 3rd ed. (Pergamon, Oxford, 1980).
  65. C. Tsallis, R. S. Mendes, and A. R. Plastino, The role of constraints within generalized nonextensive statistics, Physica (Amsterdam) 261A, 534 (1998).
  66. S. Durr, Z. Fodor, C. Hoelbling, S. D. Katz, S. Krieg, T. Kurth, L. Lellouch, T. Lippert, K. K. Szabo, and G. Vulvert (BMW Collaboration), Lattice QCD at the physical point: Light quark masses, Phys. Lett. B 701, 265 (2011).
  67. C. Patrignani et al. (Particle Data Group), Review of Particle Physics, Chin. Phys. C 40, 100001 (2016).
  68. A. Bazavov et al. (HotQCD Collaboration), Chiral crossover in QCD at zero and non-zero chemical potentials, Phys. Lett. B 795, 15 (2019).
  69. L. Adamczyk et al. (STAR Collaboration), Bulk properties of the medium produced in relativistic heavy-ion collisions from the beam energy scan program, Phys. Rev. C 96, 044904 (2017).
  70. F. Gao, J. Chen, Y.-X. Liu, S.-X. Qin, C. D. Roberts, and S. M. Schmidt, Phase diagram and thermal properties of strong-interaction matter, Phys. Rev. D 93, 094019 (2016).
  71. S. Acharya et al. (ALICE Collaboration), Multiplicity dependence of light-flavor hadron production in pp collisions at s=7  TeV, Phys. Rev. C 99, 024906 (2019).
  72. J. Adam et al. (ALICE Collaboration), Enhanced production of multi-strange hadrons in high-multiplicity proton–proton collisions, Nat. Phys. 13, 535 (2017).
  73. B. Abele et al. (ALICE Collaboration), Enhanced production of multi-strange hadrons in high-multiplicity proton–proton collisions at s=7  TeV with ALICE, Phys. Lett. B 712, 309 (2012).
  74. S. Acharya et al. (ALICE Collaboration), Multiplicity dependence of π, K, and p production in pp collisions at s=13 TeV, Eur. Phys. J. C 80, 693 (2020).
  75. S. Acharya et al. (ALICE Collaboration), Multiplicity dependence of K*(892)0 and ϕ(1020) production in pp collisions at s=13 TeV, Phys. Lett. B 807, 135501 (2020).
  76. B. Abele et al. (ALICE Collaboration), Multiplicity dependence of pion, kaon, proton and lambda production in p–Pb collisions at sNN=5.02 TeV, Phys. Lett. B 728, 25 (2014).
  77. J. Adam et al. (ALICE Collaboration), Multi-strange baryon production in p-Pb collisions at sNN=5.02 TeV, Phys. Lett. B 758, 389 (2016).
  78. J. Adam et al. (ALICE Collaboration), Production of K*(892)0 and ϕ(1020) production in p–Pb collisions at sNN=5.02 TeV, Eur. Phys. J. C 76, 245 (2016).
  79. B. Abele et al. (ALICE Collaboration), Centrality dependence of π, K and p production in Pb-Pb collisions at sNN=2.76 TeV, Phys. Rev. C 88, 044910 (2013).
  80. B. Abele et al. (ALICE Collaboration), KS0 and Λ Production in Pb-Pb collisions at sNN=2.76 TeV, Phys. Rev. Lett. 111, 222301 (2013).
  81. B. Abele et al. (ALICE Collaboration), Multi-strange baryon production at mid-rapidity in Pb–-Pb collisions at sNN=2.76  TeV, Phys. Lett. B 728, 216 (2014).
  82. B. Abele et al. (ALICE Collaboration), K*(892)0 and ϕ(1020) production in Pb-Pb collisions at sNN=2.76 TeV, Phys. Rev. C 91, 024609 (2015).
  83. S. Acharya et al. (ALICE Collaboration), Production of charged pions, kaons, and (anti-)protons in Pb-Pb and inelastic pp collisions at sNN=5.02 TeV, Phys. Rev. C 101, 044907 (2020).
  84. S. Acharya et al. (ALICE Collaboration), Production of K*(892)0 and ϕ(1020) in pp and Pb-Pb collisions at sNN=5.02 TeV, Phys. Rev. C 106, 034907 (2022).
  85. S. Acharya et al. (ALICE Collaboration), Production of pions, kaons, (anti-)protons and mesons in Xe–-Xe collisions at sNN=5.44 TeV, Eur. Phys. J. C 81, 584 (2021).
  86. F. Becattini, J. Cleymans, A. Keränen, E. Suhonen, and K. Redlich, Features of particle multiplicities and strangeness production in central heavy ion collisions between 1.7A and 158A GeV/c, Phys. Rev. C 64, 024901 (2001).
  87. A. Andronic, P. Braun-Munzinger, K. Redlich, and J. Stachel, Decoding the phase structure of QCD via particle production at high energy, Nature (London) 561, 321 (2018).
  88. F. A. Flor, G. Olinger, and R. Bellwied, Flavour and energy dependence of chemical freeze-out temperatures in relativistic heavy ion collisions from RHIC-BES to LHC energies, Phys. Lett. B 814, 136098 (2021).
  89. F. A. Flor, G. Olinger, and R. Bellwied, System size and flavour dependence of chemical freeze-out temperatures in ALICE data from pp, pPb and PbPb collisions at LHC energies, Phys. Lett. B 834, 137473 (2022).
  90. L. Adamczyk et al. (STAR Collaboration), Strange hadron production in Au+Au collisions at sNN=7.7, 11.5, 19.6, 27, and 39 GeV, Phys. Rev. C 102, 034909 (2020).
  91. B. I. Abelev et al. (STAR Collaboration), Systematic measurements of identified particle spectra in pp, d+Au, and Au+Au collisions at the STAR detector, Phys. Rev. C 79, 034909 (2009).
  92. G. Agakishiev et al. (STAR Collaboration), Strangeness Enhancement in Cu-Cu and Au-Au Collisions at sNN=200 GeV, Phys. Rev. Lett. 108, 072301 (2012).
  93. C. Tsallis, Introduction to Nonextensive Statistical Mechanics (Springer, New York, 2009).

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Phys. Rev. D 112, 096022– Published 19 November, 2025
  • Received 1 July 2025
  • Revised 24 September 2025
  • Accepted 7 October 2025
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