Hydrodynamic limit of the Vlasov-Poisson-Fokker-Planck system in low-field regime

In this paper, we study the hydrodynamic limit of the scaled Vlasov-Poisson-Fokker-Planck (VPFP) system in the low-field regime. By employing the moment method, we formally derive the corresponding Drift-Diffusion-Poisson (DDP) system. Furthermore, we rigorously justify the pointwise convergence from the VPFP system to the DDP system through delicate high-order energy estimates based on the Macro-Micro decomposition. The main difficulty lies in controlling the nonlinear coupling between the kinetic and electrostatic fields and establishing uniform bounds with respect to the scaling parameter. These challenges are overcome by developing refined high-order energy methods that yield uniform energy estimates and ensure the global well-posedness of smooth solutions, without relying on any a priori assumptions for the limiting DDP system.