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 $\mathcal{K}_{22}+4\mathcal{K}_{11}^2$.