Systematic Analysis of Flow Distributions

The information of the event-by-event fluctuations is extracted from flow harmonic distributions and cumulants, which can be done experimentally. In this work, we employ the standard method of Gram-Charlier series with the normal kernel to find such distribution, which is the generalization of recently introduced flow distributions for the studies of the event-by-event fluctuations. Also, we introduce a new set of cumulants \(j_n\{2k\}\) which have more information about the fluctuations compared with other known cumulants. The experimental data imply that not only all of the information about the event-by-event fluctuations of collision zone properties and different stages of the heavy-ion process are not encoded in the radial flow distribution \(p(v_n)\), but also the observables describing harmonic flows can generally be given by the joint distribution \(\mathcal{P}(v_1,v_2,...)\). In such a way, we first introduce a set of joint cumulants \(\mathcal{K}_{nm}\), and then we find the flow joint distribution using these joint cumulants. Finally, we show that the Symmetric Cumulants \(SC(2,3)\) and \(SC(2,4)\) obtained from ALICE data are explained by the combinations \(\mathcal{K}_{22}+\frac{1}{2}\mathcal{K}_{04}-\mathcal{K}_{31}\) and \(\mathcal{K}_{22}+4\mathcal{K}_{11}^2\).