Higher order conserved charge fluctuations inside the mixed phase

General formulas are presented for higher order cumulants of the conserved charge statistical fluctuations inside the mixed phase. As a particular example, the van der Waals model in the grand canonical ensemble is used. The higher order measures of the conserved charge fluctuations up to the hyperkurtosis are calculated in a vicinity of the critical point (CP). The analysis includes both the mixed phase region and the pure phases on the phase diagram. It is shown that even-order fluctuation measures, e.g., scaled variance, kurtosis, and hyperkurtosis, have only positive values in the mixed phase and go to infinity at the CP. For odd-order measures, such as skewness and hyperskewness, the regions of positive and negative values are found near the left and right binodals, respectively. The obtained results are discussed in a context of the event-by-event fluctuation measurements in heavy-ion collisions.

Figure 1

The $(n,T)$ phase diagram of first-order phase transition in the reduced variables $\tilde{n}=n/n_c$ and $\tilde{T}=T/T_c$.

Figure 2

The vdW model results in the reduced ($\tilde{n},\tilde{T}$) coordinates for the (a) scaled variance, (b) skewness, and (c) kurtosis. The mixed phase fluctuation values are obtained using Eq. (13). The binodals and the CP are represented by the green lines and the green points, respectively.

Figure 3

Hyperskewness and hyperkurtosis are calculated, including the $x$ fluctuation effects using Eq. (12), and shown, respectively, in panels (a) and (c). For comparison the same quantities calculated with Eq. (13), i.e., when $x$ fluctuations are neglected, are presented in panels (b) and (d).