Shear Viscosity of an $N$-Component Gas Mixture using the Chapman--Enskog Method under Anisotropic Scatterings

The analytical Chapman-Enskog formula for calculating the shear viscosity $\eta$ of a relativistic ideal gas, such as a massless quark-gluon plasma, has consistently demonstrated good agreement with the numerical results obtained using the Green-Kubo relation under both isotropic and anisotropic two-body scatterings. However, past analyses of massless, multicomponent quark-gluon plasma have focused on an effective single-component "gluon gas." The Chapman-Enskog formula for multicomponent mixtures with nonzero yet adjustable masses was previously developed for simpler cases of isotropic scatterings. This study aims to obtain the Chapman-Enskog shear viscosity formula for a massless, multicomponent mixture under general anisotropic scatterings. Since the shear viscosity depends on a linearized collision kernel, an approximation formula for the linearized collision kernel is derived under elastic and anisotropic $l+k\rightarrow l+k$ scatterings. This derived approximation agrees very well with the isotropic two-body kernels provided in previous works for both like and different species. Furthermore, for multicomponent mixtures beyond two species types, an alternative expansion method of the $N$-component Chapman-Enskog viscosity is presented. This is applied to a two-component "binary" mixture and compared with the conventional formula for binary viscosity. The agreement between the two, for interacting and noninteracting binary mixtures, varies from moderate to well.