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

TL;DR

This work probes high-order fluctuations near a critical endpoint using molecular dynamics of a classical Lennard-Jones fluid, extending prior analyses to third- and fourth-order cumulants in both coordinate- and momentum-space acceptances and to factorial cumulants. It demonstrates that near the CP, cumulants equilibrate on timescales comparable to second-order ones but suffer strong finite-size effects, with coordinate-space signals robust while proton-based observables are significantly diluted. Momentum-space signals are washed out in equilibrium but can be revived by incorporating a Bjorken-like collective flow, aligning momentum-space behavior with coordinate-space findings. The results, including subensemble acceptance corrections and scaled factorial cumulants, provide a framework for interpreting heavy-ion fluctuation measurements (e.g., RHIC-BES-II) and for identifying CP signals through multi-particle correlations sensitive to flow and conservation laws.

Abstract

We present a systematic investigation of particle number fluctuations in the crossover region near the critical endpoint 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 time scales 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, $κσ^2 \ll -1$, 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 $|κσ^2| \lesssim 1$ 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.

Figures (12)

• Figure 1: Scaled variance (a), skewness (b), and kurtosis (c) of grand-canonical particle number fluctuations in the Kolafa-Nezbeda equation of state (see Appendix \ref{['KolafaEOS']} for details) for the Lennard–Jones fluid in the $(\tilde{T},\tilde{n})$ plane. The critical point is indicated by a black star, while the binodal and spinodal lines are shown by solid gray and solid black curves, respectively. Contours of constant values for fluctuation measures are shown as thin lines. The isotherm at $T = 1.06\,T_{\rm c}$ is represented by a dashed line, and the three densities along this isotherm selected for simulations, $n = 0.32\,n_{\rm c}$, $n = 0.95\,n_{\rm c}$, and $n = 1.9\,n_{\rm c}$, are marked by red, blue, and green circles, respectively.
• Figure 2: Scaled variance, skewness, and kurtosis of grand-canonical particle-number fluctuations in the Kolafa–Nezbeda (black solid curves) and virial equation of state (yellow dash-dotted curves). Van der Waals EoS results (green dotted curves) are provided for comparison. The dots represent the maximum value for $N=400$ LJ data with GCE correction (see Fig. \ref{['fig-moments-alphadep']}).
• Figure 3: Scaled variance (a), skewness (b), and kurtosis (c) of particle-number fluctuations in the Lennard–Jones fluid within a coordinate-space subsystem with $\alpha=0.2$ at temperature $T = 1.06T_{\rm c}$ as a function of the reduced time $\tilde{t}$. Three values of density, $n=0.32 n_{\rm c}$, $n=0.95 n_{\rm c}$, and $n=1.9 n_{\rm c}$, are presented by, respectively, red, blue, and green bands. The widths of the bands represent statistical uncertainties obtained using the Delta method. The inset (d) highlights the early-time behavior of the kurtosis. The dot-dashed lines correspond to the parametric fit by exponential relaxation (\ref{['eq:fit']}) with parameter values from Tab. \ref{['tab:exp']}.
• Figure 4: The same as in Fig. \ref{['fig-timedep']} but for scaled variance (a), skewness (b), and kurtosis (c) as functions of acceptance fraction $\alpha$ after thermalization. Panels (d), (e), and (g) show the corresponding fluctuation measures corrected for global baryon number conservation using Eqs. (\ref{['eq:wtil']})-(\ref{['eq:kurtgce']}), respectively. The dash-dotted lines in (d) show the expected result in the thermodynamic limit. Additionally, panel (g) shows a zoomed-in view of the kurtosis behavior for densities far from the critical point. The dashed red lines represent the ideal gas baseline.
• Figure 5: $\alpha$-dependence of kurtosis $\kappa\sigma^2$ in the vicinity of the critical point of Lennard–Jones fluid, $T = 1.06T_{\rm c}$ and $n = 0.95n_{\rm c}$ (middle) evaluated within a coordinate-space subsystem at $\tilde{t} = 50$. Baryon-number fluctuations are shown as blue bands, while proton-number fluctuations with isospin randomization and with additional charge conservation are represented by pink and yellow bands, respectively. The corresponding ideal-gas baselines are indicated by dashed (isospin randomization) and dash-dotted (charge conservation) lines. Note that ideal gas baseline is the same for baryons and protons under charge conservation scenario.