Stress tensor and bulk viscosity in relativistic nuclear collisions

We discuss the influence of different initial conditions for the stress tensor and the effect of bulk viscosity on the expansion and cooling of the fireball created in relativistic heavy ion collisions. In particular, we explore the evolution of longitudinal and transverse components of the pressure and the extent of dissipative entropy production in the one-dimensional, boost-invariant hydrodynamic model. We find that a bulk viscosity consistent with recent estimates from lattice QCD further slows the equilibration of the system; however, it does not significantly increase the entropy produced.

Figure 1

(a) Kinematic bulk viscosity $\zeta /s$ and kinematic shear viscosity $\eta /s$ as functions of temperature $T$. $\eta /s$ is determined by the KSS bound $\eta /s=1/(4\pi )$; and $\zeta /s$ is derived by a fit to the results reported in Ref. [15], using the conformal measure ω as scaling variable. This fit is denoted as $c_{\zeta }=a_{\zeta }=1$. The blue dashed curve shows a modified parametrization for $\zeta /s$ with double the peak height ($c_{\zeta }=2$) and half the peak width ($a_{\zeta }=2$). (b) Parametrization of the reduced interaction measure $(\epsilon -3P)/T^4$ from Ref. [14] as a function of $T$.

Figure 2

(a) Relative bulk and shear stresses, $-\Pi /P$ and $\Phi /P$, as functions of time τ for the scenario (iii, S, SYM, $c_{\zeta }=1$). The time when $T_c$ is reached is indicated by the triangle. (b) Relative transverse and longitudinal pressures, $P_{\perp }/P$ and $P_z/P$, as functions of time τ for (iii, S, SYM, $c_{\zeta }=1$) (solid lines) and for the same scenario but with $c_{\zeta }$ set to zero (dashed lines). (c) Relative entropy production from bulk and shear stress, $S_{\Pi }/S_f$ and $S_{\Phi }/S_f$, as functions of time τ for the same scenarios with $c_{\zeta }=1$. All results are for $a_{\zeta }=1$.

Figure 3

(a) Relative longitudinal pressure $P_z/P$ as function of time τ for scenario (i, S, SYM) and various choices for the bulk viscosity ($c_{\zeta }=0,1,2$ and $a_{\zeta }=1,2)$. The solid line shows our “standard” parametrization ($c_{\zeta }=a_{\zeta }=1$). (b) Relative entropy production from bulk stress, $S_{\Pi }/S_f$, as function of time τ for the same scenarios as in part (a).

Figure 4

(a) Relative transverse and longitudinal pressure, $P_{\perp }/P$ and $P_z/P$, as functions of time τ for (S, SYM, $c_{\zeta }=1$) and initial conditions (i) (black), (ii) (blue), and (iii) (red). (b) Relative entropy production from bulk and shear stress, $S_{\Pi }/S_f$ and $S_{\Phi }/S_f$, as functions of time τ for the same set of scenarios. Note that the different curves for $S_{\Pi }/S_f$ lie almost on top of each other. The triangle indicates the time of critical temperature for initial condition (iii). For conditions (i) and (ii), $T_c$ is reached slightly earlier. All curves are for $a_{\zeta }=1$.

Figure 5

(a) Relative transverse and longitudinal pressure, $P_{\perp }/P$ and $P_z/P$. (b) Relative entropy production from bulk and shear stress, $S_{\Pi }/S_f$ and $S_{\Phi }/S_f$. The different scenarios shown for initial conditions (iii) are ${\tau }_{\pi }={\tau }_{\Pi }={\tau }_{\pi }^{(\mathrm{kin})}$ from kinetic theory, see Eq. (11), no conformal terms in Eq. (6), $c_{\zeta }=1$ [red]; ${\tau }_{\pi }={\tau }_{\Pi }={\tau }_{\pi }^{(\mathrm{SYM})}$ from conformal symmetry, see Eq. (12), no conformal terms in Eq. (6), $c_{\zeta }=1$ [blue]; ${\tau }_{\pi }={\tau }_{\Pi }={\tau }_{\pi }^{(\mathrm{SYM})}$, conformal terms in Eq. (6) switched on, $c_{\zeta }=1$ [black]; ${\tau }_{\pi }={\tau }_{\Pi }={\tau }_{\pi }^{(\mathrm{SYM})}$, conformal terms in Eq. (6) switched on, $c_{\zeta }=0$ [gray]. The triangles indicate the largest and smallest times of critical temperature corresponding to the last and the first scenarios, respectively.