On two possible ways to recover ordinary thermodynamics from extended thermodynamics of polyatomic gases

We consider two possible ways, i.e. the Maxwellian iteration (MI) and the Chapman-Enskog method (CEM), to recover relativistic ordinary thermodynamics from relativistic extended thermodynamics of Polyatomic gases with N moments. Both of these methods give the Eckart equations which are the relativistic version of the Navier-Stokes and Fourier laws as a first iteration. However, these methods do not lead to the same expressions of the heat conductivity chi, the shear viscosity mu, and the bulk viscosity nu which appear as coefficients in the Eckart equations. In particular, we prove that the expressions of chi, mu, and nu obtained via the CEM do not depend on N, while those obtained through the MI depend on N. Moreover, we also prove that these two methods lead to the same results in the nonrelativistic limit.