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.

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In 2011, a macroscopic extended model with many moments for the description of relativistic gases under the influence of an external electromagnetic field has been proposed. An exact closure of this model, up to whatever order with respect to thermodynamical equilibrium, has been found. Such closure is given in terms of the entropy density at equilibrium and the expression for the higher order terms involves a family of arbitrary constants. It allows to know the value of the constitutive functions up to every desired degree of approximation, but paying the cost of a large amount of arbitrary terms. Moreover, if one wants to use the closure for applications it is necessary to cut it at a chosen order with respect to thermodynamical equilibrium. In this paper, we propose a different method to find the closure that, starting from a fixed order with respect to thermodynamical equilibrium, allows to obtain univocally all the lower order terms in terms of two arbitrary functions depending on a scalar variable and of one polynomial which degree depends on the order of approximation.

The many moments model for dense gases and macromolecular fluids is considered here, where the upper order moment is chosen in accordance to the suggestions of the non-relativistic limit of the corresponding relativistic model. The solutions of the restrictions imposed by the entropy principle and that of Galilean relativity were, until now, obtained in the literature by using Taylor expansions around equilibrium and without proving convergence. Here, an exact solution without using expansions is found. The particular case with only 14 moments has already been treated in the literature in a completely different way. Here, it is proven that this particular closure is included in the presently more general one.

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