Transport coefficients of second-order relativistic fluid dynamics in the relaxation-time approximation

TL;DR

This paper derives and compares second-order relativistic fluid transport coefficients using two methods, resolving discrepancies and validating results through numerical solutions, with implications for accurate modeling of relativistic fluids.

Contribution

It provides a consistent derivation of second-order transport coefficients using the method of moments and Chapman-Enskog in the relaxation-time approximation, clarifying previous conflicting results.

Findings

Transport coefficients derived are in perfect agreement between methods.

Identifies divergence issues in diffusion-shear coupling coefficients at high expansion order.

Validates theoretical results with numerical solutions of the Boltzmann equation.

Abstract

We derive the transport coefficients of second-order fluid dynamics with $14$ dynamical moments using the method of moments and the Chapman-Enskog method in the relaxation-time approximation for the collision integral of the relativistic Boltzmann equation. Contrary to results previously reported in the literature, we find that the second-order transport coefficients derived using the two methods are in perfect agreement. Furthermore, we show that, unlike in the case of binary hard-sphere interactions, the diffusion-shear coupling coefficients $\ell_{V\pi}$, $\lambda_{V\pi}$, and $\tau_{V\pi}$ actually diverge in some approximations when the expansion order $N_\ell \rightarrow \infty$. Here we show how to circumvent such a problem in multiple ways, recovering the correct transport coefficients of second-order fluid dynamics with $14$ dynamical moments. We also validate our results for the diffusion-shear coupling by comparison to a numerical solution of the Boltzmann equation for the propagation of sound waves in an ultrarelativistic ideal gas.