  A reduced drift-diffusion (Smoluchowski-Poisson) equation is found for the
electric charge in the high-field limit of the Vlasov-Poisson-Fokker-Planck
system, both in one and three dimensions. The corresponding electric field
satisfies a Burgers equation. Three methods are compared in the one-dimensional
case: Hilbert expansion, Chapman-Enskog procedure and closure of the hierarchy
of equations for the moments of the probability density. Of these methods, only
the Chapman-Enskog method is able to systematically yield reduced equations
containing terms of different order.
