Abstract: We consider the one dimensional Euler-Poisson system, in the { \em isothermal case}, with a {\em friction coefficient} $\epsi^{-1}.$ When $\epsi \rightarrow 0_+$, we show that the sequence of entropy-admissible weak solutions constructed in \cite{PRV} converges to the solution to the drift-diffusion equations. We use the scaling introduced in \cite{MN2}, who proved a quite similar result in the {\em isentropic} case, using the theory of compensated compactness. On one hand this theory cannot be used in our case; on the other hand, exploiting the linear pressure law, we can give here a much simpler proof by only using the entropy inequality and de la Vall\'ee-Poussin criterion of weak compactness in $L^1.$
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