Shear viscosity, Bulk viscosity and Relaxation Times of Causal Dissipative Relativistic Fluid-Dynamics at Finite Temperature and Chemical Potential

TL;DR

This paper derives microscopic formulas for viscosities and relaxation times in causal relativistic fluid dynamics at finite temperature and chemical potential, highlighting the importance of operator definitions and the TCL approximation.

Contribution

It provides a consistent derivation of transport coefficients at finite chemical potential using the projection operator method and discusses the physical implications of the TCL approximation.

Findings

The ratio of bulk viscosity to relaxation time matches the zero chemical potential case when operator definitions are chosen properly.

The TCL approximation correctly breaks time-reversal symmetry and aligns with quantum master equation results.

Next-order corrections to TCL can reproduce previous results but introduce issues.

Abstract

The microscopic formulas for the shear viscosity $\eta$, the bulk viscosity $\zeta$, and the corresponding relaxation times $\tau_\pi$ and $\tau_\Pi$ of causal dissipative relativistic fluid-dynamics are obtained at finite temperature and chemical potential by using the projection operator method. The non-triviality of the finite chemical potential calculation is attributed to the arbitrariness of the operator definition for the bulk viscous pressure.We show that, when the operator definition for the bulk viscous pressure $\Pi$ is appropriately chosen, the leading-order result of the ratio, $\zeta$ over $\tau_\Pi$, coincides with the same ratio obtained at vanishing chemical potential. We further discuss the physical meaning of the time-convolutionless (TCL) approximation to the memory function, which is adopted to derive the main formulas. We show that the TCL approximation violates the time reversal symmetry appropriately and leads results consistent with the quantum master equation obtained by van Hove. Furthermore, this approximation can reproduce an exact relation for transport coefficients obtained by using the f-sum rule derived by Kadanoff and Martin. Our approach can reproduce also the result in Baier et al.(2008) Ref. \cite{con} by taking into account the next-order correction to the TCL approximation, although this correction causes several problems.