Study of transport coefficients in ultrarelativistic kinetic theory

A spatially-periodic longitudinal wave is considered in relativistic dissipative hydrodynamics. At sufficiently small wave amplitudes, an analytic solution is obtained in the linearised limit of the macroscopic conservation equations within the first- and second-order relativistic hydrodynamics formulations. A kinetic solver is used to obtain the numerical solution of the relativistic Boltzmann equation for massless particles in the Anderson-Witting approximation for the collision term. It is found that, at small values of the Anderson-Witting relaxation time $\tau$, the transport coefficients emerging from the relativistic Boltzmann equation agree with those predicted through the Chapman-Enskog procedure, while the relaxation times of the heat flux and shear pressure are equal to $\tau$. These claims are further strengthened by considering a moment-type approximation based on orthogonal polynomials under which the Chapman-Enskog results for the transport coefficients are exactly recovered.