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

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