# Study of transport coefficients in ultrarelativistic kinetic theory

## TL;DR

This paper investigates transport coefficients in relativistic kinetic theory using analytical solutions, numerical Boltzmann equation solutions, and polynomial approximations, confirming consistency with Chapman-Enskog predictions at small relaxation times.

## Contribution

It provides a comprehensive analysis of transport coefficients in relativistic hydrodynamics, combining analytical, numerical, and approximation methods to validate theoretical predictions.

## Key findings

- Transport coefficients agree with Chapman-Enskog predictions at small relaxation times.
- Numerical solutions of the Boltzmann equation confirm analytical results.
- Orthogonal polynomial approximations recover Chapman-Enskog results exactly.

## Abstract

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.