Bulk Viscosity and Relaxation Time of Causal Dissipative Relativistic Fluid Dynamics

TL;DR

This paper derives microscopic formulas for bulk viscosity and relaxation time in causal relativistic fluid dynamics, applying them to pionic fluids and comparing with other theoretical approaches to ensure causality.

Contribution

It provides new microscopic formulae for bulk viscosity and relaxation time using the projection operator method, ensuring consistency with causality in relativistic fluids.

Findings

Relaxation time is enhanced near the QCD phase transition.

Derived formulas are consistent with Israel-Stewart and string theory results.

The formulas satisfy the causality condition in relativistic fluid dynamics.

Abstract

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 $% \tau_{\Pi}$ 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.