Shear viscosity of nuclear matter

Shear viscosity $\eta$ is calculated for the nuclear matter described as a system of interacting nucleons with the van der Waals (VDW) equation of state. The Boltzmann-Vlasov kinetic equation is solved in terms of the plane waves of the collective overdamped motion. In the frequent-collision regime, the shear viscosity depends on the particle-number density $n$ through the mean-field parameter $a$, which describes attractive forces in the VDW equation. In the temperature region $T=15–40$ MeV, a ratio of the shear viscosity to the entropy density $s$ is smaller than 1 at the nucleon number density $n=(0.5–1.5)n_0^{}$, where $n_0^{}=0.16{\mathrm{fm}}^{-3}$ is the particle density of equilibrium nuclear matter at zero temperature. A minimum of the $\eta /s$ ratio takes place somewhere in a vicinity of the critical point of the VDW system. Large values of $\eta /s\gg 1$ are, however, found in both the low-density, $n\ll n_0^{}$, and high-density, $n>2n_0^{}$, regions. This makes the ideal hydrodynamic approach inapplicable for these densities.

Figure 1

The geometry of the collision of two hard spheres in the center mass system; $\mathbf{q}={\mathbf{p}}_1-\mathbf{p},{\mathbf{q}}^{\prime }={\mathbf{p}}_1^{\prime }-{\mathbf{p}}^{\prime },O{O}_1=d,d$ is the sphere particle diameter, $\beta$ is the impact parameter; ${\theta }_{p^{\prime }}$ is the scattering angle.

Figure 2

Shear viscosity $\eta$ [Eq. (36)] in the frequent-collision regime in units of the CE value ${\eta }_{\mathrm{CE}}^{}$ [Eq. (1)] versus particle-number density $n$ in units of the normal density $n_0=0.16{\mathrm{fm}}^{-3}$ of nuclear matter; $m\cong 938$ MeV; $d=1\mathrm{fm},a=100\mathrm{MeV}{\mathrm{fm}}^3$.

Figure 3

Contour plot for the ratio $\eta /s$ of the DPW viscosity $\eta$ [Eq. (36)] to the entropy density $s$ (5) in the $n\text{−}T$ plane at $g=4$ and the same $m,d$, and $a$ parameters as in Fig. 2.

Figure 4

Same as in Fig. 3 but for the ratio of the hydrodynamic CE shear viscosity [Eq. (1)] to the entropy density [Eq. (5)].

Figure 5

Integral $y^{\lambda }{\mathcal{J}}_{St}\left(y\right)$ [Eq. (A13), solid] vs $y$ for powers $\lambda =2$ (red), 3 (blue), and 4 (black); dashed lines show those with the corresponding asymptotics [Eq. (A15)] up to fourth order.