  A kinetic consideration of an axisymmetric equilibrium with vanishing
electric field near the magnetic axis shows that del f should not vanish on
axis within the framework of Vlasov theory while it can either vanish or not in
the framework of both a drift kinetic and a gyrokinetic theories (f is either
the pertinent particle or the guiding center distribution function). This
different behavior, relating to the reduction of phase space which leads to the
loss of a Vlasov constant of motion, may result in the construction of
different currents in the reduced phase space than the Vlasov ones. This
conclusion is indicative of some limitation on the implications of reduced
kinetic theories in particular as concerns the physics of energetic particles
in the central region of magnetically confined plasmas.
