Traces of nonequilibrium dynamics in relativistic heavy-ion collisions

The impact of nonequilibrium effects on the dynamics of heavy-ion collisions is investigated by comparing a nonequilibrium transport approach, the Parton-Hadron-String-Dynamics (PHSD), to a 2D+1 viscous hydrodynamical model, which is based on the assumption of local equilibrium and conservation laws. Starting the hydrodynamical model from the same nonequilibrium initial condition as in the PHSD, using an equivalent lQCD equation of state (EoS), the same transport coefficients, i.e., shear viscosity $\eta$ and the bulk viscosity $\zeta$ in the hydrodynamical model, we compare the time evolution of the system in terms of energy density, Fourier transformed energy density, spatial and momentum eccentricities, and ellipticity to quantify the traces of nonequilibrium phenomena. In addition, we also investigate the role of initial pre-equilibrium flow on the hydrodynamical evolution and demonstrate its importance for final state observables. We find that because of nonequilibrium effects, the event-by-event transport calculations show large fluctuations in the collective properties, while ensemble averaged observables are close to the hydrodynamical results.

Figure 1

Initial conditions for hydrodynamics: the energy density profiles from a single PHSD event (a) and averaged over 100 PHSD events (b) taken at $t=0.6$ fm/c for a peripheral ($b=6$ fm) Au+Au collision at $\sqrt{s_{NN}}=200$ GeV.

Figure 2

Initial conditions of a peripheral Au+Au collision ($b$=6 fm) at $\sqrt{s_{NN}}=200$ GeV (calculated within the PHSD). (Top contour) The local energy density in the transverse plane; (middle contour) quiver plot of initial flow $v_x,v_y$ in the transverse plane (color map indicates the magnitude of local initial flow); (bottom contour) initial flow with respect to the distance of the cell from the center of energy density.

Figure 3

$\eta /s$ and $\zeta /s$ versus scaled temperature $T/T_C$. (a) The symbols indicate the PHSD results of $\eta /s$ from Ref. [24], calculated using different methods: the relaxation-time approximation (red line+diamonds) and the Kubo formalism (blue line+dots); the black line corresponds to the parametrization of the PHSD results for $\eta /s$. The orange short dashed line demonstrates the Kovtun-Son-Starinets bound [43] ${(\eta /s)}_{\mathrm{KSS}}=1/\left(4\pi \right).$ The orange dashed line is the $\eta /s$ of VISHNU hydrodynamical model that was recently determined by Bayesian analysis. (b) $\zeta /s$ from PHSD simulations from Ref. [24] and the $\zeta /s$ that is adapted in our hydrodynamical simulations.

Figure 4

Evolution of the ratio of the transverse $P_T$ and longitudinal $P_L$ pressures over the cell energy density $e$ extracted from PHSD as a function of time for different cells along the $x$ axis in a peripheral ($b=6$ fm) Au+Au collision at $\sqrt{s_{NN}}=200$ GeV. Note that $T^{\mu \nu }$ was averaged over 100 PHSD events.

Figure 5

Evolution of the relative value between the transverse pressure extracted from PHSD $P_T$ (see also Fig. 4 for the pressure components) and the pressure $P$ given by the EoS for different times in the transverse plane of a peripheral ($b=6$ fm) Au+Au collision at $\sqrt{s_{NN}}=200$ GeV. Note that $T^{\mu \nu }$ was averaged over 100 PHSD events.

Figure 6

Local energy density $e(x,y,z=0)$ (left column) and the corresponding temperature $T(x,y,z=0)$ given by the EoS (right column) in the transverse plane from a PHSD event (NUM = 30) at different times for a peripheral ($b$ = 6 fm) Au+Au collision at $\sqrt{s_{NN}}=200$ GeV.

Figure 7

Local energy density $e(x,y,z=0)$ (left column) and the corresponding temperature $T(x,y,z=0)$ given by the EoS (right column) in the transverse plane from a single hydrodynamical event at different proper times for a peripheral ($b$ = 6 fm) Au+Au collision at $\sqrt{s_{NN}}=200$ GeV.

Figure 8

Contour plots of the local energy density $e$ in the transverse plane from the PHSD simulation of one event, for a peripheral Au+Au collision ($b=6$ fm) at $\sqrt{s_{NN}}=200$ GeV.

Figure 9

Contour plots of the local energy density $e$ in the transverse plane from the hydrodynamical simulation starting from the same initial conditions as in Fig. 8 including initial flow.

Figure 10

Components of the 3-velocity (${\beta }_x,{\beta }_y,{\beta }_z$) in the transverse plane from a single PHSD event (NUM = 30) at a different proper time for a peripheral ($b$ = 6 fm) Au+Au collision at $\sqrt{s_{NN}}=200$ GeV. ${\beta }_z$ is scaled differently from ${\beta }_x,{\beta }_y$ for better orientation.

Figure 11

Components of the velocity (${\beta }_x,{\beta }_y$) in the transverse plane from a single hydrodynamical event (taking the same initial condition as the PHSD event above) at a different proper time for a peripheral ($b$ = 6 fm) Au+Au collision at $\sqrt{s_{NN}}=200$ GeV.

Figure 12

Contour plots of the Fourier transform of the energy density $\tilde{e}(x,y,z=0)$ for the simulation obtained within PHSD as shown in Fig. 8.

Figure 13

Contour plots of the Fourier transform of the energy density $\tilde{e}(x,y,z=0)$ for the simulation obtained within hydrodynamics as shown in Fig. 9.

Figure 14

Radial distribution of the Fourier modes of the energy density for a different proper time in both PHSD and hydrodynamical events. The red lines correspond to the PHSD simulations and the black line corresponds to the hydrodynamical simulations.

Figure 15

Event-by-event averaged spatial eccentricity ${\epsilon }_2$ of 100 PHSD events and 100 VISHNU events with respect to proper time, for a peripheral Au+Au collision ($b=6$ fm) at $\sqrt{s_{NN}}=200$ GeV. The green dots show the distribution of each of the 100 PHSD events used in this analysis. The solid red line is the average over all the green dots. The blue, yellow, and black lines correspond to hydrodynamical evolution taking different initial condition scenarios.

Figure 16

Event-by-event averaged total momentum anisotropy of 100 PHSD events and 100 VISHNU events with respect to proper time, for a peripheral Au+Au collision ($b=6$ fm) at $\sqrt{s_{NN}}=200$ GeV. (a) The total momentum eccentricity of hydrodynamical evolution for different initial scenarios, as well as different bulk viscosity adapted in the hydrodynamical simulation. (b) Comparison of the total momentum eccentricity from PHSD events compared with the standard hydrodynamical events. The green dots show the distribution of each of the 100 PHSD events used in this analysis. The solid red line is an average over the green dots. The black line corresponds to the standard hydrodynamical evolution taking the 100 initial conditions which are generated from PHSD events.