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

TL;DR

This paper derives a Chapman-Enskog shear viscosity formula for a massless, multicomponent gas mixture under anisotropic scatterings, extending previous isotropic models and providing approximation methods validated against known kernels.

Contribution

It develops a generalized Chapman-Enskog shear viscosity formula for multicomponent, massless gases with anisotropic scatterings, including an approximation for the collision kernel and an alternative expansion method.

Findings

The approximation formula agrees well with previous isotropic kernels.

The alternative expansion method works for mixtures beyond two species.

Binary mixture viscosity calculations show moderate to good agreement.

Abstract

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.