Transport coefficients of second-order relativistic fluid dynamics in the relaxation-time approximation
Victor E. Ambruş, E. Molnár, Dirk H. Rischke
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 ${\ensuremath{\ell}}_{V\ensuremath{\pi}}$, ${\ensuremath{\lambda}}_{V\ensuremath{\pi}}$, and ${\ensuremath{\tau}}_{V\ensuremath{\pi}}$ actually diverge in some approximations when the expansion order ${N}_{\ensuremath{\ell}}\ensuremath{\rightarrow}\ensuremath{\infty}$. 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.