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

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 $N_{\ell }\to \infty$. 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.

22 pages and 2 figures, minor corrections and updated references

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