Effect of initial fluctuations on the collective flow in intermediate-energy heavy ion collisions

A systemical analysis of the initial fluctuation effect on the collective flows for $\text{Au}+\text{Au}$ at $1A$ GeV is presented in the framework of isospin-dependent quantum molecular dynamics model (IQMD), and a special focus on the initial fluctuation effect on the squeeze out is emphasized. The flows calculated by the participant plane reconstructed by the initial geometry in coordinate space are compared with those calculated by both the ideal reaction-plane and event-plane methods. It is found that initial fluctuation weakens the squeeze-out effect, and some discrepancies between the flows extracted by these different plane methods appear, which indicate that the flows are affected by the evolution of dynamics. In addition, we find that the squeeze-out flow is also proportional to initial eccentricity. Our calculations also qualitatively give a trend for the excitation function of the elliptic flow similar to that of the FOPI Collaboration experimental data. Finally, we address the nucleon number scaling of the flows for light particles. Even though the initial fluctuation significantly decreases the ratio $v_4/v_2^2$ and $v_3/(v_1{v}_2)$, all fragments to mass number 4 keep the same curve and show independence from transverse momentum.

Figure 1

[(a), (b)] Comparison for directed flow $v_1$, [(c), (d)] elliptic flow $v_2$, and [(e), (f)] triangular flow $v_3$ with respect to reaction plane (red dots), participant plane (blue up triangles) and event plane (green down triangles) in $\text{Au}+\text{Au}$ collisions at $E=1A$ GeV and impact-parameter zone $0.25<b_0<0.45$ from the IQMD simulation with the SM parameter. The black stars are the experimental data from FOPI (note the $v_3$ data of FOPI are not available). The $y_0$ dependence integrated over $u_{t_0}$ but constrained to $u_{t_0}>0.4$ is plotted in the left panel. The right panel shows the$u_{t_0}$ dependence in the indicated $y_0$ region.

Figure 2

(upper panels) Rapidity dependence of directed flow $v_1$ and (lower panels) elliptic flow $v_2$ of protons in $\text{Au}+\text{Au}$ collisions for different indicated centrality ranges. Transverse four-velocities $u_{t_0}$ below 0.4 are cut off.

Figure 3

Average elliptic flow with respect to participant plane, $\langle v_2^{\text{PP}}\rangle$, as a function of participant-plane eccentricity ${\varepsilon }_2$ at four given impact-parameter ranges for $\text{Au}+\text{Au}$ collisions at $E=1A$ GeV. The line describes a proportional relation of the absolute values of $v_2$ versus ${\varepsilon }_2$ with a slope of $-0.045$.

Figure 4

Average density evolutions with time at different impact parameters. A central sphere with radius 5 fm is selected to calculate the density. The maximum compression is around 9 fm/$c$.

Figure 5

Average ${\varepsilon }_2$ evolutions with time at different impact parameters. The red dash line is around 9 fm/$c$.

Figure 6

Elliptic flow with respect to different reaction planes as a function of beam energy for $Z=1$ particles in $\text{Au}+\text{Au}$ collisions at impact parameters of $5.5\text{fm}<b<7.5\text{fm}$. The solid squares represent the experimental data [39] and different lines are for different reaction-plane methods.

Figure 7

(left panels) Comparison between directed flow $v_1$ and elliptic flow $v_2$ from IQMD-SM simulation and (right panels) IQMD-HM simulation for $\text{Au}+\text{Au}$ collisions at $E=1A$ GeV in $0.25<b_0<0.45$ centrality range. $u_{t_0}\text{below}$ 0.4 are cut off.

Figure 8

(a) Rapidity dependence of the $C\left(A\right)$ scaled $v_1$ and (b) $v_2$ for light particles of $\text{Au}+\text{Au}$ at $1A$ GeV. Different symbols represent different light fragments. The lines are just to guide the eyes.

Figure 9

(a) Ratios of $v_4/v_2^2$ and (b) $v_3/(v_1{v}_2)$ as a function of transverse momentum $p_T$ for $\text{Au}+\text{Au}$ at $1A$ GeV. Different symbols represent different fragments as indicted in inset. Solid symbols represent the ratios when the flows are calculated without initial fluctuation. Open symbols for the ratios when the initial fluctuation is taken into account for flow calculations. The lines are just to guide the eyes.