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A reduced drift-diffusion (Smoluchowski-Poisson) equation is found for theelectric charge in the high-field limit of the Vlasov-Poisson-Fokker-Plancksystem, both in one and three dimensions. The corresponding electric fieldsatisfies a Burgers equation. Three methods are compared in the one-dimensionalcase: Hilbert expansion, Chapman-Enskog procedure and closure of the hierarchyof equations for the moments of the probability density. Of these methods, onlythe Chapman-Enskog method is able to systematically yield reduced equationscontaining terms of different order.

HIGH-FIELD LIMIT OF THE VLASOV–POISSON–FOKKER–PLANCK SYSTEM: A COMPARISON OF DIFFERENT PERTURBATION METHODS