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We present a brief history of how the famous Maximum Entropy Principle was used as closure of moments of the Boltzmann equation. In particular, we want to remark on the important role of two fundamental papers by Wolfgang Dreyer, one in the classical framework and one in a relativistic context, to use this principle and to compare the result with the macroscopic theory of Rational Extended Thermodynamics., Comment: 28 pages, 0 figures, Paper accepted to a Volume dedicated to the memory of Wolfgang Dreyer (Springer 2025)

A novel ES-BGK-based model of non-polytropic rarefied gases in the framework of kinetic theory is presented. Key features of this model are: an internal state density function depending only on the microscopic energy of internal modes (avoiding the dependence on temperature seen in previous reference studies); full compliance with the H-theorem; feasibility of the closure of the system of moment equations based on the maximum entropy principle, following the well-established procedure of Rational Extended Thermodynamics. The structure of planar shock waves in carbon dioxide (CO$_2$) obtained with the present model is in general good agreement with that of previous results, except for the computed internal temperature profile, which is qualitatively different with respect to the results obtained in previous studies, showing here a consistently monotonous behavior across the shock structure, rather than the non monotonous behavior previously found.

We present a complete analysis of the dispersion relations for longitudinal and transverse waves in a rarefied polyatomic gas based on Rational Extended Thermodynamics (RET), which describes the evolution of a non-polytropic gas in nonequilibrium. Observability of the second mode of the longitudinal wave and the transverse wave is discussed although these waves are usually not payed much attention. The cases of CO$_2$ gas and para-H$_2$ gas are specifically analyzed as typical examples., Comment: 12 pages. Submitted to RIVISTA DI MATEMATICA DELLA UNIVERSITA' DI PARMA

A relativistic version of the Kinetic Theory for polyatomic gas is considered and a new hierarchy of moments that takes into account the total energy composed by the rest energy and the energy of the molecular internal modes is presented. In the first part, we prove via classical limit that the truncated system of moments dictates a precise hierarchy of moments in the classical framework. In the second part, we consider the particular physical case of fifteen moments closed via Maximum Entropy Principle in a neighborhood of equilibrium state. We prove that this symmetric hyperbolic system satisfies all the general assumption of some theorems that guarantee the global existence of smooth solutions for initial data sufficiently small.

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.

Motivated by a recent paper of Pennisi (2021) in the relativistic framework, we revisit the previous approach of two hierarchies of moments critically and propose a new natural physical hierarchy of moments to describe classical rarefied non-polytropic polyatomic gas in the framework of Molecular Rational Extended Thermodynamics. The differential system of the previous approach is proved to be a \emph{principal subsystem} of the present one. The main idea is that at the molecular level, the total energy is the kinetic energy plus the energy of internal mode due to the rotation and vibration, and the increasing moments contain this total energy as power in which the power index increases with the number of tensorial indexes. In particular, we consider the case of $15$ moments, and we close the system using the variational method of the Maximum Entropy Principle. We prove the convexity of entropy and the possibility to put the system in symmetric form. This more rich kinetic framework may be interesting also as possible applications to biomathematics or other fields in which kinetic models were applied recently with success., Comment: 28 pages, 2 figures

Recently, Pennisi and Ruggeri [J Stat Phys 179, 231-246 (2020)] consider the classical limit of the relativistic theory of moments associated with the Boltzmann-Chernikov equation truncated at a tensorial index $N+1$ and they proved that there exists a unique possible choice of the moments in the classical case for a given $N$ both for monatomic and polyatomic gases. In particular, in polyatomic gases, there exists a new hierarchy of moments that is more general than the one considered in the recent literature. As consequence, when $N=2$, in the classical limit, there is a theory with $15$ fields. In this paper, we consider this system of moments, and we close the system using the maximum entropy principle. It is shown that the theory contains as a principal subsystem the previously polyatomic $14$ fields theory, and in the monatomic limit, in which the dynamical pressure vanishes, the differential system converges instead to Grad 13-moments system to the 14 moments theory proposed by Kremer [Annales de l'I.H.P. Physique th\'eorique, 45, 419-440 (1986)]., Comment: 21 pages, 2 figures. This is the pre-peer reviewed version of the following article: MDPI journal Fluids

We present a more refined version of rational extended thermodynamics of rarefied polyatomic gases in which molecular rotational and vibrational relaxation processes are treated individually. In this case we need a triple hierarchy of the moment system and the system of balance equations is closed via the maximum entropy principle. Three different types of the production terms in the system, which are suggested by a generalized BGK-type collision term in the Boltzmann equation, are adopted. In particular, the rational extended thermodynamic theory with seven independent fields (ET7) is analyzed in detail. Finally, the dispersion relation of ultrasonic wave derived from the ET7 theory is confirmed by the experimental data for CO2, Cl2, and Br2 gases., Comment: 20 pages, 8 figures, original paper -