  This work explores the different shapes that can be realized by the
one-particle velocity distribution functions (VDFs) associated with the
fourth-order maximum-entropy moment method. These distributions take the form
of an exponential of a polynomial of the particle velocity, with terms up to
the fourth-order. The 14- and 21-moment approximations are investigated.
Various non-equilibrium gas states are probed throughout moment space. The
resulting maximum-entropy distributions deviate strongly from the equilibrium
VDF, and show a number of lobes and branches. The Maxwellian and the
anisotropic Gaussian distributions are recovered as special cases. The
eigenvalues associated with the maximum-entropy system of transport equations
are also illustrated for some selected gas states. Anisotropic and/or
asymmetric non-equilibrium states are seen to be associated with a non-uniform
spacial propagation of perturbations.
