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The microscopic formulae of the bulk viscosity $\zeta $ and the correspondingrelaxation time $\tau_{\Pi}$ in causal dissipative relativistic fluid dynamicsare derived by using the projection operator method. In applying these formulaeto the pionic fluid, we find that the renormalizable energy-momentum tensorshould be employed to obtain consistent results. In the leading orderapproximation in the chiral perturbation theory, the relaxation time isenhanced near the QCD phase transition and $\tau_{\Pi}$ and $\zeta $ arerelated 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, pressureand velocity of sound, respectively. The predicted $\zeta $ and $% \tau_{\Pi}$should satisfy the so-called causality condition. We compare our result withthe results of the kinetic calculation by Israel and Stewart and the stringtheory, and confirm that all the three approaches are consistent with thecausality condition.

Bulk Viscosity and Relaxation Time of Causal Dissipative Relativistic Fluid Dynamics