Shear viscosity of a binary mixture for a relativistic fluid at high temperature

The determination of the shear viscosity is a central topic in various areas of modern physics. In particular, it is often necessary to evaluate the shear viscosity $η$ of fluids made up of more than one species, all interacting with different cross sections. Since it may be difficult to extract information on the interaction among different species, various combinations of the viscosities of the individual components are often used. We work in the Chapman-Enskog framework and investigate on binary mixtures, by comparing such single component combinations with a full 2-component formalism: we find that, in most cases, the full viscosity is well approximated by a weighted linear average of the single component viscosities, although this result is far from being general. Moreover, we validate our 2-component Chapman-Enskog results for $η$ by comparing them with an independent numerical simulation of the Boltzmann equation, which estimates the shear viscosity via a Green-Kubo formula, in the case of a quasi-particle system that reproduces lattice QCD thermodynamics. We see that the temperature dependence of $η/s$ of such system of quarks and gluons is not well described by combinations of the individual components, highlighting the importance of inter-species scattering.

💡 Research Summary

The paper investigates the shear viscosity η of a relativistic fluid composed of two distinct particle species at high temperature, using the Chapman‑Enskog (CE) expansion of the Boltzmann equation. The authors first review the CE formalism for a single component gas, where the first‑order approximation yields η⁻¹ = (10 T γ₀²)/c₀₀, with γ₀ and c₀₀ expressed through relativistic omega integrals that depend on the particle mass‑to‑temperature ratio (z = m/T) and the differential cross‑section σ(θ). They then extend the formalism to binary mixtures, deriving a compact expression (Eq. 7) that involves the masses, concentrations, and a set of transport coefficients c_{kl}. The diagonal coefficients c_{kk} contain contributions from same‑species collisions (σ_{11}, σ_{22}), while the off‑diagonal term c_{12} encodes inter‑species scattering (σ_{12}) and includes factors such as the reduced mass μ₁₂, total mass M₁₂, and a temperature scale ζ₁₂ = M₁₂/(2T).

To assess the practical usefulness of simpler approximations, the authors compare the full two‑component CE result with four widely used single‑component interpolation formulas: (i) linear sum η = η₁ + η₂, (ii) weighted linear average η = x₁η₁ + x₂η₂ (with x_k the particle fractions), (iii) inverse average η =