  We derive a four-component Vlasov equation for a system composed of spin-1/2
fermions (typically electrons). The orbital part of the motion is classical,
whereas the spin degrees of freedom are treated in a completely
quantum-mechanical way. The corresponding hydrodynamic equations are derived by
taking velocity moments of the phase-space distribution function. This
hydrodynamic model is closed using a maximum entropy principle in the case of
three or four constraints on the fluid moments, both for Maxwell-Boltzmann and
Fermi-Dirac statistics.
