Shear viscosity of a strongly interacting system: Green-Kubo correlator versus Chapman-Enskog and relaxation-time approximations

The shear viscosity $\eta$ has been calculated by using the Green-Kubo relation in the framework of a partonic transport approach solved at cascade level. We compare the numerical results for $\eta$ obtained from the Green-Kubo correlator with the analytical formulas in both the relaxation time approximation (RTA) and the Chapman-Enskog (CE) approximation. We investigate and emphasize the differences between the isotropic and anisotropic cross sections and between the massless and massive particles. We show that in the range of temperature explored in a heavy ion collision and for perturbative-QCD-like cross section, the RTA significantly underestimates the viscosity by about a factor of 2–3, while a good agreement is found between the CE approximation and Gree-Kubo relation already at first-order of approximation. The agreement with the CE approximation supplies an analytical formula that allows us to develop a kinetic transport theory at fixed shear-viscosity to entropy-density ratio, $\eta /s$. This opens the possibility of exploring dissipative nonequilibrium evolution of the distribution functions versus $T$-dependent $\eta /s$ and particle momenta in the dynamics of the quark-gluon plasma created in ultrarelativistic heavy-ion collisions.

Figure 1

Correlation function $\langle {\pi }^{xy}\left(t\right){\pi }^{xy}\left(0\right)\rangle$ as a function of time for different values of the total cross section ${\sigma }_{\mathrm{tot}}$; the temperature is fixed to $T=0.4$ GeV. These results are for massless particles and for isotropic cross section.

Figure 2

Left: Relaxation time $\tau$ as a function of the maximum time $T_{\max }$. Right: Initial value of the correlator $\langle {\pi }^{xy}\left(0\right){\pi }^{xy}\left(0\right)\rangle$ as a function of $T_{\max }$. The total cross section is fixed to ${\sigma }_{\mathrm{tot}}=0.1{\mathrm{fm}}^2$ and the temperature is $T=0.4\mathrm{GeV}$. The dashed line is the analytical result.

Figure 3

Left: Relaxation time $\tau$ as a function of the maximum time $N_{\mathrm{test}}$. Right: Initial value of the correlator $\langle {\pi }^{xy}\left(0\right){\pi }^{xy}\left(0\right)\rangle$ as a function of $N_{\mathrm{test}}$. The total cross section is fixed to ${\sigma }_{\mathrm{tot}}=0.1$ fm${}^2$ and the temperature is $T=0.4$ GeV. The dashed line is the analytical result.

Figure 4

Shear viscosity $\eta$ for a massless system as a function of the total isotropic cross section $\sigma$ and for different temperatures. The circles are the results from the Green-Kubo method, while the lines are the results obtained using the Chapman-Enskog approximation in Eq. (21).

Figure 5

$\eta /s$ as a function of the temperature and for different cross sections ${\sigma }_{\mathrm{tot}}=1$, 3, 6, 9, and 30 mb. The circles are the results of the Green-Kubo method while the lines are the results obtained using Eq. (24) from the CE approximation.

Figure 6

Left: Shear viscosity $\eta$ as a function of the Debye mass $m_D$ for three different values of the temperature $T=0.3,0.4,0.5\mathrm{GeV}$ blue, green and red respectively. The open simbols are the results obtained using the Green-Kubo relation. The solid, dashed and dot dashed lines refer to the RTA approach with ${\tau }^{-1}=\langle \rho {\sigma }_{\mathrm{tr}}{v}_{\mathrm{rel}}\rangle$ respectively for $T=0.3,0.4,0.5\mathrm{GeV}$. The dotted line is the isotropic limit when $m_D\to \infty$. Right: The same as the left panel but the solid, dashed and dot dashed lines refer to the Chapman-Enskog approximation at first order.

Figure 7

Left: Shear viscosity $\eta$ as a function of the Debye mass $m_D$ for three different values of the temperature $T=0.3$ GeV (blue solid lines), $T=0.4$ GeV (red dashed lines), and $T=0.5$ GeV (green dash-dotted lines). The dotted line is the isotropic limit when $m_D\gg 1$ GeV, the lines are the relaxation time approximation with ${\tau }^{-1}=\langle \rho {\sigma }_{\mathrm{tr}}{v}_{\mathrm{rel}}\rangle$, Right: The lines represent the Chapman-Enskog approximation at first order for the three different temperatures ($T=0.3$, 0.4, and 0.5 GeV) while the open circles are the results obtained using the Green-Kubo relation.

Figure 8

Shear-viscosity to entropy-density ratio $\eta /s$ of a gluon plasma interacting through the differential cross section in Eq. (39) as a function of temperature $T$; red circles are the results obtained using Green-Kubo correlator, and the red dashed line represents the Chapman-Enskog first-order approximation. The dot-dashed line is the prediction with the modified RTA approximation and the dotted line is the standard RTA.