  We establish Extended Thermodynamics (ET) of rarefied polyatomic gases with
six independent fields, i.e., the mass density, the velocity, the temperature
and the dynamic pressure, without adopting the near-equilibrium approximation.
The closure is accomplished by the Maximum Entropy Principle (MEP) adopting a
distribution function that takes into account the internal degrees of freedom
of a molecule. The distribution function is not necessarily near equilibrium.
The result is in perfect agreement with the phenomenological ET theory. To my
knowledge, this is the first example of molecular extended thermodynamics with
a non-linear closure. The integrability condition of the moments requires that
the dynamical pressure should be bounded from below and from above. In this
domain the system is symmetric hyperbolic. Finally we verify the K-condition
for this model and show the existence of global smooth solutions.
