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One of the main applications in plasma physics concerns the energy produc- tion through thermo-nuclear fusion. The controlled fusion requires the conne- ment of the plasma into a bounded domain and for this we appeal to the mag- netic connement. Several models exist for describing the evolution of strongly magnetized plasmas. The subject matter of this paper is to provide a rigorous derivation of the guiding-center approximation in the general three dimensional setting under the action of large stationary inhomogeneous magnetic elds. The rst order corrections are computed as well : electric cross eld drift, magnetic gradient drift, magnetic curvature drift, etc. The mathematical analysis relies on average techniques and ergodicity.

Gyrokinetic Vlasov Equation in Three Dimensional Setting. Second Order Approximation