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

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 toward local equilibrium ${f}_{0\mathbf{k}}$, by replacing ${f}_{\mathbf{k}}\ensuremath{-}{f}_{0\mathbf{k}}$ via ${f}_{\mathbf{k}}\ensuremath{-}{f}_{\mathrm{S}\mathbf{k}}$. The Shakhov-like distribution ${f}_{\mathrm{S}\mathbf{k}}$ is constructed using ${f}_{0\mathbf{k}}$ and the irreducible moments ${\ensuremath{\rho}}_{r}^{{\ensuremath{\mu}}_{1}\ensuremath{\cdots}{\ensuremath{\mu}}_{\ensuremath{\ell}}}$ 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 ${\ensuremath{\lambda}}_{\mathrm{\ensuremath{\Pi}}\ensuremath{\pi}}$ in the frame of the Bjorken flow of massive particles, as well as the diffusion-shear transport coefficients ${\ensuremath{\ell}}_{V\ensuremath{\pi}}$, ${\ensuremath{\ell}}_{\ensuremath{\pi}V}$ 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.