  The genesis of the ion axial velocity distribution function (VDF) is analyzed
for collisionless Hall thruster discharges. An analytical form for the VDF is
obtained from the Vlasov equation, by applying the Tonks-Langmuir theory in the
thruster channel, under the simplifying assumptions of monoenergetic creation
of ions and steady state. The equivalent set of 1D unsteady anisotropic moment
equations is derived from the Vlasov equation, and simple phenomenological
closures are formulated, assuming a polynomial shape for the ions VDF. The
analytical results and the anisotropic moment equations are compared to
collisionless PIC simulations, employing either a zero heat flux (Euler-like
equations) or the polynomial-VDF closure for the heat flux. The analytical ion
VDF and its moments are then compared to experimental measurements.
