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High-order cumulants and correlation functions near the critical point from molecular dynamics

Volodymyr A. Kuznietsov1,2, Roman Poberezhniuk1,2, Mark I. Gorenstein2,3, Volker Koch4, and Volodymyr Vovchenko1

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Phys. Rev. C 113, 045209 – Published 14 April, 2026

DOI: https://doi.org/10.1103/y9jn-p5c1

Abstract

We present a systematic investigation of particle-number fluctuations in the crossover region near the critical end point of a first-order phase transition using molecular dynamics simulations of the classical Lennard-Jones fluid. We extend our prior studies to third- and fourth-order cumulants in both coordinate- and momentum-space acceptances and integrated correlation functions (factorial cumulants). We find that, even near the critical point, non-Gaussian cumulants equilibrate on timescales comparable to those of the second-order cumulants, but show stronger finite-size effects. The presence of interactions and of the critical point leads to strong deviations of the cumulants from the ideal-gas baseline in coordinate space; these deviations are expected to persist in momentum space in the presence of collective expansion. In particular, the kurtosis becomes strongly negative, κσ2≪−1, on the crossover side of the critical point. However, this signal is significantly diluted once an efficiency cut used to distinguish protons from baryons is applied, leading to |κσ2|≲1 even in the presence of the critical point. We discuss our results in the context of ongoing measurements of proton-number cumulants in heavy-ion collisions in RHIC-BES-II.

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References (40)

  1. L. D. Landau and E. M. Lifshitz, Statistical Physics, 3rd ed. (Butterworth-Heinemann, Oxford, England, 1996).
  2. A. Bzdak, S. Esumi, V. Koch, J. Liao, M. Stephanov, and N. Xu, Mapping the phases of quantum chromodynamics with beam energy scan, Phys. Rep. 853, 1 (2020).
  3. M. A. Stephanov, K. Rajagopal, and E. V. Shuryak, Event-by-event fluctuations in heavy ion collisions and the QCD critical point, Phys. Rev. D 60, 114028 (1999).
  4. Y. Hatta and M. A. Stephanov, Proton-number fluctuation as a signal of the QCD critical end point, Phys. Rev. Lett. 91, 102003 (2003); Publisher's Note: Proton-number fluctuation as a signal of the QCD critical end point [Phys. Rev. Lett. 91, 102003 (2003)], 91, 129901 (2003) .
  5. M. A. Stephanov, Non-Gaussian fluctuations near the qcd critical point, Phys. Rev. Lett. 102, 032301 (2009).
  6. M. Stephanov, QCD critical point: Recent developments, EPJ Web Conf. 314, 00042 (2024).
  7. M. S. Abdallah, et al. (STAR Collaboration), Measurements of proton high-order cumulants in sNN=3GeVAu+Au collisions and implications for the qcd critical point, Phys. Rev. Lett. 128, 202303 (2022).
  8. B. E. Aboona, et al. (STAR Collaboration), Precision measurement of net-proton-number fluctuations in Au+Au collisions at RHIC, Phys. Rev. Lett. 135, 142301 (2025).
  9. M. Abdallah, et al. (STAR Collaboration), Cumulants and correlation functions of net-proton, proton, and antiproton multiplicity distributions in Au+Au collisions at energies available at the BNL Relativistic Heavy Ion Collider, Phys. Rev. C 104, 024902 (2021).
  10. V. Koch, Hadronic fluctuations and correlations, in Relativistic Heavy Ion Physics, edited by R. Stock (Springer, Berlin, 2010), pp. 626–652.
  11. B. Berdnikov and K. Rajagopal, Slowing out of equilibrium near the QCD critical point, Phys. Rev. D 61, 105017 (2000).
  12. O. Savchuk, R. V. Poberezhnyuk, A. Motornenko, J. Steinheimer, M. I. Gorenstein, and V. Vovchenko, Phase transition amplification of proton number fluctuations in nuclear collisions from a transport model approach, Phys. Rev. C 107, 024913 (2023).
  13. V. Vovchenko, V. Koch, and C. Shen, Proton number cumulants and correlation functions in Au-Au collisions at sNN=7.7–200 GeV from hydrodynamics, Phys. Rev. C 105, 014904 (2022).
  14. V. A. Kuznietsov, O. Savchuk, M. I. Gorenstein, V. Koch, and V. Vovchenko, Critical point particle number fluctuations from molecular dynamics, Phys. Rev. C 105, 044903 (2022).
  15. V. A. Kuznietsov, O. Savchuk, R. V. Poberezhnyuk, V. Vovchenko, M. I. Gorenstein, and H. Stoecker, Molecular dynamics analysis of particle number fluctuations in the mixed phase of a first-order phase transition, Phys. Rev. C 107, 055206 (2023).
  16. A. Bzdak, V. Koch, and V. Vovchenko, Acceptance dependence of factorial cumulants, long-range correlations, and the antiproton puzzle, Phys. Rev. C 112, 024901 (2025).
  17. V. Kuznietsov and V. Vovchenko, High-order cumulants and correlation functions near the critical point from molecular dynamics, Zenodo, (2025), http://doi.org/10.5281/zenodo.17575258.
  18. S. Stephan, J. Staubach, and H. Hasse, Review and comparison of equations of state for the Lennard-Jones fluid, Fluid Phase Equilib. 523, 112772 (2020).
  19. S. Stephan, M. Thol, J. Vrabec, and H. Hasse, Thermophysical properties of the Lennard-Jones fluid: Database and data assessment, J. Chem. Inf. Model. 59, 4248 (2019).
  20. J. Kolafa and I. Nezbeda, The Lennard-Jones fluid: an accurate analytic and theoretically-based equation of state, Fluid Phase Equilib. 100, 1 (1994).
  21. V. Vovchenko, D. V. Anchishkin, and M. I. Gorenstein, Particle number fluctuations for the van der Waals equation of state, J. Phys. A: Math. Theor. 48, 305001 (2015).
  22. V. Vovchenko, R. V. Poberezhnyuk, D. V. Anchishkin, and M. I. Gorenstein, Non-Gaussian particle number fluctuations in vicinity of the critical point for van der Waals equation of state, J. Phys. A: Math. Theor. 49, 015003 (2016).
  23. M. P. Allen and D. J. Tildesley, Computer Simulation of Liquids (Oxford University Press, Oxford, UK, 2017).
  24. V. Vovchenko, Molecular dynamics simulation and visualization of the Lennard-Jones system utilizing CUDA-enabled GPUs (2023), https://github.com/vlvovch/lennard-jones-cuda.
  25. V. A. Kuznietsov, M. I. Gorenstein, V. Koch, and V. Vovchenko, Coordinate versus momentum cuts and effects of collective flow on critical fluctuations, Phys. Rev. C 110, 015206 (2024).
  26. V. Vovchenko, A header-only C++ library for mean and standard error estimation of moments and cumulants (2022), https://github.com/vlvovch/sample-moments.
  27. M. G. Kendall and A. Stuart, The Advanced Theory of Statistics, Volume 1: Distribution Theory, 4th ed. (Macmillan, London, 1977).
  28. A. W. van der Vaart, Asymptotic Statistics (Cambridge University Press, Cambridge, UK, 1998).
  29. X. Luo, Unified description of efficiency correction and error estimation for moments of conserved quantities in heavy-ion collisions, Phys. Rev. C 91, 034907 (2015); Erratum: Unified description of efficiency correction and error estimation for moments of conserved quantities in heavy-ion collisions [Phys. Rev. C 91, 034907 (2015)], 94, 059901 (2016).
  30. V. Vovchenko, O. Savchuk, R. V. Poberezhnyuk, M. I. Gorenstein, and V. Koch, Connecting fluctuation measurements in heavy-ion collisions with the grand-canonical susceptibilities, Phys. Lett. B 811, 135868 (2020).
  31. A. Bzdak, V. Koch, and N. Strodthoff, Cumulants and correlation functions versus the QCD phase diagram, Phys. Rev. C 95, 054906 (2017).
  32. J. Adam, et al. (STAR Collaboration), Nonmonotonic energy dependence of net-proton number fluctuations, Phys. Rev. Lett. 126, 092301 (2021).
  33. C. Chattopadhyay, J. Ott, T. Schaefer, and V. V. Skokov, Simulations of stochastic fluid dynamics near a critical point in the phase diagram, Phys. Rev. Lett. 133, 032301 (2024).
  34. R. V. Poberezhnyuk, O. Savchuk, M. I. Gorenstein, V. Vovchenko, K. Taradiy, V. V. Begun, L. Satarov, J. Steinheimer, and H. Stoecker, Critical point fluctuations: Finite size and global charge conservation effects, Phys. Rev. C 102, 024908 (2020).
  35. M. A. Stephanov, Sign of kurtosis near the qcd critical point, Phys. Rev. Lett. 107, 052301 (2011).
  36. M. Kitazawa and M. Asakawa, Relation between baryon number fluctuations and experimentally observed proton number fluctuations in relativistic heavy ion collisions, Phys. Rev. C 86, 024904 (2012); Erratum: Relation between baryon number fluctuations and experimentally observed proton number fluctuations in relativistic heavy ion collisions [Phys. Rev. C 86, 024904 (2012)], 86, 069902 (2012) .
  37. M. Gottschalk, An EOS for the Lennard-Jones fluid: A virial expansion approach, AIP Adv. 9, 125206 (2019).
  38. O. Savchuk, R. V. Poberezhnyuk, V. Vovchenko, and M. I. Gorenstein, Binomial acceptance corrections for particle number distributions in high-energy reactions, Phys. Rev. C 101, 024917 (2020).
  39. V. Vovchenko, R. V. Poberezhnyuk, and V. Koch, Cumulants of multiple conserved charges and global conservation laws, J. High Energy Phys. 10 (2020) 089.
  40. R. B. Potts, Note on the factorial moments of standard distributions, Aust. J. Phys. 6, 498 (1953).

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Phys. Rev. C 113, 045209– Published 14 April, 2026
  • Received 10 November 2025
  • Accepted 19 February 2026
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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.

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