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For the one-species Vlasov-Poisson-Boltzmann (VPB) system in the scalingunder which the moments of the fluctuations formally converge to theincompressible Navier-Stokes-Fourier-Poisson (NSFP) system, we prove theuniform estimates with respect to the Knudsen number $\epsilon$ for thefluctuations. As a consequence, the existence of the global-in-time classicalsolutions of VPB with all $\epsilon \in (0,1]$ is established in whole spaceunder small size of initial data, and the convergence to incompressible NSFP as$\epsilon$ go to 0 is rigorously justified.

From Vlasov-Poisson-Boltzmann system to incompressible Navier-Stokes-Fourier-Poisson system: convergence for classical solutions