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The Massless Electron limit of the Vlasov-Poisson-Landau system

Due to ion-electron collisions, it is impossible to derive any two-fluid model for plasma as a direct hydrodynamic limit of the Vlasov-Poisson-Landau system for ions and electrons. At the same time, electrons are much lighter than their ion counterparts. In this work, we derive the massless electron limit of the Vlasov-Poisson-Landau system. This is done via a re-scaling of the electron velocity, leading to multiple velocity scales. Importantly, we demonstrate that ion-electron collisions vanish in this limit, due to special structure of the Landau collisions. We also show that this is invalid for the classical Boltzmann kernel with hard sphere interaction. This mechanism serves as the first step for the derivation of the two-fluid model for ions from a two-species kinetic equation.