Bulk Viscosity and Relaxation Time of Causal Dissipative Relativistic Fluid Dynamics

The microscopic formulae of the bulk viscosity $\zeta$ and the corresponding relaxation time $\tau_{\Pi}$ in causal dissipative relativistic fluid dynamics are derived by using the projection operator method. In applying these formulae to the pionic fluid, we find that the renormalizable energy-momentum tensor should be employed to obtain consistent results. In the leading order approximation in the chiral perturbation theory, the relaxation time is enhanced near the QCD phase transition and $\tau_{\Pi}$ and $\zeta$ are related as $\tau_{\Pi}=\zeta /[\beta \{(1/3-c_{s}^{2})(\epsilon +P)-2(\epsilon -3P)/9\}]$, where $\epsilon$, $P$ and $c_{s}$ are the energy density, pressure and velocity of sound, respectively. The predicted $\zeta$ and $$ should satisfy the so-called causality condition. We compare our result with the results of the kinetic calculation by Israel and Stewart and the string theory, and confirm that all the three approaches are consistent with the causality condition.