Extended Thermodynamics of Dense Gases with an arbitrary but fixed number of moments has been recently studied in literature; the arbitrariness of the number of moments is linked to a number N and the resulting model is called an (N ) − M odel. As usual in Extended Thermodynamics, in the field equations some unknown functions appear; restriction on their generalities are obtained by imposing the entropy principle, the Galilean relativity principle and some symmetry conditions. The solution of these conditions is called the "closure problem" and it has not been written explicitly because an hard notation is necessary, but it has been shown how the theory is selfgenerating in the sense that, if we know the closure of the (N ) − M odel, than we will be able to find that of the (N + 1) − M odel. Instead of this, we find here an exact solution which holds for every number N .

A new model for Polyatomic Gases with an arbitrary but fixed number of moments has been recently proposed and investigated in the framework of Extended Thermodynamics; the arbitrariness of the number of moments is linked to a numberNand the resulting model is called anN-Model. This model has been elaborated in order to take into account the entropy principle, the Galilean relativity principle, and some symmetry conditions. It has been proved that the solution for all these conditions exists, but it has not been written explicitly because hard notation is necessary; it has only been shown how the theory is self-generating in the sense that if we know the closure of theN-Model, then we will be able to find that of(N+1)-Model. Up to now only a single particular solution has been found in this regard. Instead of this, we find here a numberable set of exact solutions which hold for every fixed numberN.

Extended Thermodynamics of dense gases is characterized by two hierarchies of field equations, which allow one to overcome some restrictions on the generality of the previous models. This idea has been introduced by Arima, Taniguchi, Ruggeri and Sugiyama. In the case of a 14-moment model, they have found the closure of the balance equations up to second order with respect to equilibrium. Here, the closure is obtained up to whatever order and imposing only the necessary symmetry conditions. It comes out that the first non-symmetric parts of the higher order fluxes appear only at third order with respect to equilibrium, even if Arima, Taniguchi, Ruggeri and Sugiyama found a non-symmetric part proportional to an arbitrary constant also at first order with respect to equilibrium. Consequently, this constant must be zero, as Arima, Taniguchi, Ruggeri and Sugiyama assumed in the applications and on an intuitive ground.

A new model for Polyatomic and for Dense Gases has been proposed in literature in the last five years in the framework of Extended Thermodynamics. The case with an arbitrary but fixed number of moments has been recently studied, both with the kinetic approach than with the macroscopic approach; this last one is more general and includes the results of the kinetic approach only as a particular case.Scope of the "closure problem" is to find the expression of some arbitrary functions which appear in the balance equations. Up to now only a recurrence procedure has been published which outlines how to find the solution of this problem with the macroscopic approach; by using this procedure, a numberable set of solutions has been found and written explicitly, while we find here the most general exact solution. It is determined except for some arbirary terms and it is interesting that these terms appear also in the 24 moments model; so we find here that they are transmitted from the model with 24 moments to those with an arbitrary number of moments, without any further arbitrary term.

Maxwell’s equations in materials are studied jointly with Euler equations using new knowledge recently appeared in the literature for polyatomic gases. For this purpose, a supplementary conservation law is imposed; one of the results is a restriction on the law linking the magnetic field in empty space and the electric field in materials to the densities of the 4-Lorentz force να and its dual μα: These are the derivatives of a scalar function with respect to να and μα, respectively. Obviously, two of Maxwell’s equations are not evolutive (Gauss’s magnetic and electric laws); to simplify this mathematical problem, a new symmetric hyperbolic set of equations is here presented which uses unconstrained variables and the solutions of the new set of equations, with initial conditions satisfying the constraints, restore the previous constrained set. This is also useful because it allows to maintain an overt covariance that would be lost if the constraints were considered from the beginning. This is also useful because in this way the whole set of equations becomes a symmetric hyperbolic system as usually in Extended Thermodynamics.