DISCRETE AND CONTINUOUS Website: http://AIMsciences.org DYNAMICAL SYSTEMS–SERIES B Volume 4, Number 4, November 2004 pp. 1129–1142 DIFFUSION APPROXIMATION FOR THE ONE DIMENSIONAL BOLTZMANN-POISSON SYSTEM N. Ben Abdallah Math´ ematiques pour l’Industrie et la Physique, UMR, CNRS 5640 Universit´ e Paul Sabatier 118 route de Narbonne, 31062 Toulouse, France M. Lazhar Tayeb Laboratoire d’Ing´ eniere Math´ ematique Ecole Polytechnique de Tunisie La Marsa, Tunisie (Communicated by Benoˆ ıt Perthame) Abstract. The diffusion limit of the initial-boundary value problem for the Boltzmann-Poisson system is studied in one dimension. By carefully analyzing entropy production terms due to the boundary, L p estimates are established for the solution of the scaled Boltzmann equation (coupled to Poisson) with well prepared initial and boundary conditions. A hybrid Hilbert expansion taking advantage of the regularity of the limiting system allows to prove the convergence of the solution towards the solution of the Drift-Diffusion-Poisson system and to exhibit a convergence rate. 1. Introduction and main result. In kinetic transport theory, particles are de- scribed by their time-dependent distribution function f (x, v, t) defined on the phase space (position-velocity) and solution of a Boltzamnn type equation. This equa- tion provides a good description of the transport phenomena in many applications like plasmas, semiconductors, rarefied gases, neutron transport or bilological sys- tems. However, its dependence on several variables (position and velocity variables) makes its use in numerical simulation very expensive. Macroscopic models, widely used in applications [21, 27, 28, 29], are well suited for numerical computations. Their derivation from kinetic models has been the subject of a large amount of mathematical work. Depending on the collision phenomena, various fluid limits like diffusion lim- its, hydrodynamic or high field limits allow to derive macroscopic models from the Boltzmann equation [3, 4, 6, 7, 10, 13, 16, 17, 20, 26], etc. The diffusion limit of the Boltzmann equation with the linear BGK collision operator (modelling electron- phonon interaction) has been done by F. Poupaud in [25], where he shows the convergence towards the Drift-Diffusion equation. The analysis is done in a regular and bounded position domain with inflow boundary conditions for the Boltzmann 1991 Mathematics Subject Classification. 35B45, 35B25, 82B21, 54C70. Key words and phrases. Transport equations, Boltzmann-Poisson system, Drift-Diffusion equa- tions, entropy inequality, Hilbert expansion. 1129