Entropy principle for the moment systems of degree $\alpha$ associated to the Boltzmann equation. Critical derivatives and non controllable boundary data

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moments and of degree $\alpha (ET_m^\alpha)$. For such theories the entropy principle is still valid only, if the non equilibrium field variables and their derivatives are sufficiently small with respect to the required approximation order. In this paper we prove through simple examples of stationary problems that the entropy principle fails in general, if all the non-equilibrium variables are of the same order of magnitude. This is due to the fact that there exist some derivatives of non equilibrium variables (critical derivatives) that are not small along all the solutions. This property can be used to fix the non controllable boundary data in such a manner that the critical derivatives are kept small for the solution that we may choose. Thus, for the stationary unidimensional case we propose the requirement that the critical derivatives vanish on the boundary, eventually with some successive derivatives. This is a sufficient condition for the validity of the entropy principle at least in a neighborhood of the boundary and makes it possible to assign the non controllable data in a simple manner when the number of moments is greater than 13. We have tested this procedure in several cases of $ET_m^\alpha$ theories, showing that the criterion implies continuity with respect to the change of the moment number and to the truncation order. In particular for the planar unidimensional heat conduction problem we have obtained a behavior for the temperature that is always the same as the one predicted by the classical Fourier law. This result is in evident contrast with the minimax principle expectation. However we have qualitative differences between the temperature behavior described by Extended Thermodynamics and the one by Fourier-Navier-Stokes theory for heat conduction in radial symmetry.