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We study the diffusion limit of the Vlasov-Poisson-Fokker-Planck System. Here, we generalize the local in time results and the two dimensional results of Poupaud-Soler [F. Poupaud and J. Soler, Math. Models Methods Appl. Sci., 10(7), 1027-1045, 2000] and Goudon [T. Goudon, Math. Models Methods Appl. Sci., 15(5), 737-752, 2005] to the case of several space dimensions. Renormalization techniques, the method of moments and a velocity averaging lemma are used to prove the convergence of free energydoi:10.4310/cms.2010.v8.n2.a9 fatcat:elu7q36525fuxa5ieigthihwma