High-order Shakhov-like extension of the relaxation time approximation in relativistic kinetic theory

Abstract

In this paper we present a relativistic Shakhov-type generalization of the Anderson-Witting relaxation time model for the Boltzmann collision integral. The extension is performed by modifying the path on which the distribution function $f_{\mathbf{k}}$ is taken towards local equilibrium $f_{0\mathbf{k}}$, by replacing $f_{\mathbf{k}}-f_{0\mathbf{k}}$ via $f_{\mathbf{k}}-f_{\mathrm{S}\mathbf{k}}$. The Shakhov-like distribution $f_{\mathrm{S}\mathbf{k}}$ is constructed using $f_{0\mathbf{k}}$ and the irreducible moments ${\rho }_r^{{\mu }_1\cdots {\mu }_{\ell }}$ of $f_{\mathbf{k}}$ and reduces to $f_{0\mathbf{k}}$ in local equilibrium. Employing the method of moments, we derive systematic high-order Shakhov extensions that allow both the first- and the second-order transport coefficients to be controlled independently of each other. We illustrate the capabilities of the formalism by tweaking the shear-bulk coupling coefficient ${\lambda }_{\Pi \pi }$ in the frame of the Bjorken flow of massive particles, as well as the diffusion-shear transport coefficients ${\ell }_{V\pi }$, ${\ell }_{\pi V}$ in the frame of sound wave propagation in an ultrarelativistic gas. Finally, we illustrate the importance of second-order transport coefficients by comparison with the results of the stochastic BAMPS method in the context of the one-dimensional Riemann problem.

Comments: 38 pages, 8 figures, 2 tables