We propose a possible generalization of the BGK collisional term in the relativistic kinetic theory and we compare our model with previous ones. The present model has the advantage in that it satisfies the conservation of particle number, energy-momentum and the H-theorem in the Eckart frame without constraints on particular local equilibrium state. In the last part of the paper we extend the model to include also the polyatomic gas.

A relativistic version of the rational extended thermodynamics of polyatomic gases based on a new hierarchy of moments that takes into account the total energy composed by the rest energy and the energy of the molecular internal mode is proposed. The moment equations associated with the Boltzmann–Chernikov equation are derived, and the system for the first 15 equations is closed by the procedure of the maximum entropy principle and by using an appropriate BGK model for the collisional term. The entropy principle with a convex entropy density is proved in a neighborhood of equilibrium state, and, as a consequence, the system is symmetric hyperbolic and the Cauchy problem is well-posed. The ultra-relativistic and classical limits are also studied. The theories with 14 and 6 moments are deduced as principal subsystems. Particularly interesting is the subsystem with 6 fields in which the dissipation is only due to the dynamical pressure. This simplified model can be very useful when bulk viscosity is dominant and might be important in cosmological problems. Using the Maxwellian iteration, we obtain the parabolic limit, and the heat conductivity, shear viscosity, and bulk viscosity are deduced and plotted.

Recently Pennisi and Ruggeri [Ann. Phys. 377, 414 (2017)] proposed a casual hyperbolic model for a dissipative relativistic gas with internal structure. In this paper, we consider the particular case of the model when dissipation is negligible (Eulerian gas). We study in particular the energy behavior in comparison with the Synge energy which is valid for monatomic gas and we evaluate the characteristic velocities proving the hyperbolicity of the differential system. The second part of the paper is devoted to the ultra-relativistic limit of the model and we prove that there exists a critical value of the degree of freedom such that for smaller values of this quantity the ultra-relativistic limit of the energy of a gas with structure is the same as the Synge energy, while for larger degrees of freedom the energy increases with the degree of freedom itself.