WU Yuanfang, LI Xiaobing, CHEN Lizhu, LI Zhiming, XU Mingmei, PAN Xue, ZHANG Fan, ZHANG Yanhua, and ZHONG Yuming

The goal of relativistic heavy-ion collisions is to determine the phase boundary of quantum chromodynamics (QCD) phase transitions. Critically sensitive observables are suggested to be higher-order cumulants of conserved charges. The non-monotonous behavior of higher cumulants was observed at the relativistic heavy-ion collider (RHIC). However, it remains unclear whether these non-monotonous behaviors are critically related. We studied the influences of non-critical fluctuations, finite system size, and limited evolution time to determine if they cause non-monotonous behavior. First, we examined the minimum statistics required for measuring the fourth cumulant. The minimum statistic obtained using the centrality bin width correction (CBWC) method was 25 M. We suggest using a 0.1% centrality bin in the CBWC method instead of each Nch. With a 0.1 centrality bin width, 1 M statistics are sufficient. We then pointed out the statistical fluctuations from the limited number of final particles. By assuming the independent emission of each positive (or negative) charged particle, the statistical fluctuations of positive (or negative) charged particles were presented by a Poisson distribution, and the statistical fluctuations of net-charged particles were their evolution. The obtained statistical fluctuations for net protons, net electronic charges, and net baryons were consistent with those from the Hadron Resonance Gas model. In addition, the measured cumulants at RHIC/STAR are dominated by these Poisson-like statistical fluctuations. At the end of this section, we suggest the pooling method of mixed events and demonstrate that the sample of mixed events accurately presents the contributions of the background. Dynamic cumulants were defined as the cumulant of the original sample minus that of the mixed sample. Dynamical cumulants were shown to simultaneously reduce the influence of the statistical fluctuations, centrality bin width effects, and detector efficiency. Second, because the system is finite, the correlation length at the critical point is not developed to infinity in contrast to the system at thermal limits. Using a Monte Carlo simulation of the three-dimensional three-state Potts model, we demonstrated the fluctuations of the second- and fourth-order generalized susceptibilities near the temperatures of the external fields of the first-, second-, and crossover regions. Both the second- and fourth-order susceptibilities showed similar peak-like and oscillation-like fluctuations in the three regions. Therefore, non-monotonic fluctuations are associated with the second-order phase transition and the first-order phase and crossover in a finite-size system. The exponent of finite-size scaling (FSS) characterizes the order of transitions or crossover. To determine the parameters of the phase transition using the FSS, we studied the behavior of a fixed point in the FSS. To quantify the behavior of the fixed point, we define the width of the scaled observables of different sizes at a given temperature and scaling exponent ratio. The minimum width reveals the position of the fixed point in the plane of the temperature and scaling exponent ratio. The value of this ratio indicates the nature of the fixed point, which can be a critical, first-order phase transition line point, or crossover region point. To demonstrate the effectiveness of this method, we applied it to three typical samples produced by a three-dimensional three-state Potts model. The results show that the method is more precise and effective than conventional methods. Possible applications of the proposed method are also discussed. Finally, because of the limited evolution time, some processes in relativistic heavy-ion collisions may not reach thermal equilibrium. To estimate the influence of the nonequilibrium evolution, we used the three-dimensional Ising model with the Metropolis algorithm to study the evolution from nonequilibrium to equilibrium on the phase boundary. The order parameter exponentially approaches its equilibrium value, as suggested by the Langevin equation. The average relaxation time is defined. The relaxation time is well represented by the average relaxation time, which diverges as the zth power of the system size at a critical temperature, similar to the relaxation time in dynamical equations. During nonequilibrium evolution, the third and fourth cumulants of the order parameter could be positive or negative depending on the observation time, which is consistent with the calculations of dynamical models at the crossover side. The nonequilibrium evolution at the crossover side lasts briefly, and its influence is weaker than that at the first-order phase transition line. These qualitative features are instructive for experimentally determining the critical point and phase boundary in quantum chromodynamics.