Semiclassical Vlasov and fluid models for an electron gas with spin   effects
  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. 
Jerome Hurst, Omar Morandi, Giovanni Manfredi, Paul-Antoine Hervieux
physics.plasm-ph cond-mat.str-el quant-ph