Shear viscosity of an ultrarelativistic Boltzmann gas with isotropic inelastic scattering processes

We derive an analytic expression for the shear viscosity of an ultra-relativistic gas in presence of both elastic $2\to 2$ and inelastic $2\leftrightarrow 3$ processes with isotropic differential cross sections. The derivation is based on the entropy principle and Grad’s approximation for the off-equilibrium distribution function. The obtained formula relates the shear viscosity coefficient $\eta$ to the total cross sections ${\sigma }_{22}$ and ${\sigma }_{23}$ of the elastic resp. inelastic processes. The values of shear viscosity extracted using the Green-Kubo formula from kinetic transport calculations are shown to be in excellent agreement with the analytic results which demonstrates the validity of the derived formula.