Shear viscosity coefficient from microscopic models

The transport coefficient of shear viscosity is studied for a hadron matter through microscopic transport model, the ultrarelativistic quantum molecular dynamics (UrQMD), using the Green-Kubo formulas. Molecular-dynamical simulations are performed for a system of light mesons in a box with periodic boundary conditions. Starting from an initial state composed of $\pi ,\eta ,\omega ,\rho ,\varphi$ with a uniform phase-space distribution, the evolution takes place through elastic collisions, production, and annihilation. The system approaches a stationary state of mesons and their resonances, which is characterized by common temperature. After equilibration, thermodynamic quantities such as the energy density, particle density, and pressure are calculated. From such an equilibrated state the shear viscosity coefficient is calculated from the fluctuations of stress tensor around equilibrium using Green-Kubo relations. We do our simulations here at zero net baryon density so that the equilibration times depend on the energy density. We do not include hadron strings as degrees of freedom so as to maintain detailed balance. Hence we do not get the saturation of temperature but this leads to longer equilibration times.

Figure 1

The time evolution of particle densities for each particle with $V=1000{\text{fm}}^3$ and $\epsilon =0.3{\text{GeV}∕\text{fm}}^3$.

Figure 2

The time evolution of particle densities for each particle with $V=1000{\text{fm}}^3$ and $\epsilon =0.9\text{GeV}∕{\text{fm}}^3$.

Figure 3

Energy distributions of $\pi$, $\eta$, $\rho$, and $K$ at four different values of time, (a) $t=100\text{fm}∕c$, (b) $t=200\text{fm}∕c$, (c) $t=300\text{fm}∕c$, and (d) $t=400\text{fm}∕c$. The lines are the fitted results that are given by Boltzmann distributions, $C\exp (-\beta E)$. The calculation was done with $V=1000{\text{fm}}^3$, $n_B=0.0{\text{fm}}^{-3}$, and $\epsilon =0.9\text{GeV}∕{\text{fm}}^3$.

Figure 4

The time evolution of the inverse slopes ${\beta }^{-1}$ for (a–c) $\pi$, (a) $\eta$, (b) $\rho$, and (c) $K$ with $V=1000{\text{fm}}^3$, $n_B=0.0{\text{fm}}^{-3}$, and $\epsilon =0.9\text{GeV}∕{\text{fm}}^3$. The value of ${\beta }^{-1}$ was calculated from the fitting of energy distributions. Here the solid curves represent the time evolution of ${\beta }^{-1}$ for $\pi$.

Figure 5

The energy density of mesons as function of the temperature. The curve corresponds to the free gas model represented by Eq. (7).

Figure 6

The pressure of pions is plotted as function of the temperature. The curve corresponds to the free gas model represented by Eq. (8).

Figure 7

Stress-tensor correlation of the mesons as a function of time for different energies (a) $\epsilon =0.2\text{GeV}∕{\text{fm}}^3$, (b) $\epsilon =0.4\text{GeV}∕{\text{fm}}^3$, (c) $\epsilon =0.6\text{GeV}∕{\text{fm}}^3$, and (d) $\epsilon =0.8\text{GeV}∕{\text{fm}}^3$. The curves are the exponential fits to extract relaxation times.

Figure 8

Shear viscosity of meson gas from UrQMD simulations (data points) and of pion gas only from variational method (curve) as a function of temperature.

Figure 9

The relaxation time for the shear flux of meson gas from UrQMD (data points) and of pion gas only (curve) using shear viscosity coefficient from variational approach as a function of temperature.