High-field limit of the Vlasov-Poisson-Fokker-Planck system: A comparison of different perturbation methods

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.