Abstract
We rigorously derive a linear kinetic equation of Fokker–Planck type for a 2-D Lorentz gas in which the obstacles are randomly distributed. Each obstacle of the Lorentz gas generates a potential ε α V(\(\left( {\frac{{\left| x \right|}}{\varepsilon }} \right)\)), where V is a smooth radially symmetric function with compact support, and α>0. The density of obstacles diverges as ε −δ, where δ>0. We prove that when 0< α<1/8 and δ=2α+1, the probability density of a test particle converges as ε→0 to a solution of our kinetic equation.
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Desvillettes, L., Ricci, V. A Rigorous Derivation of a Linear Kinetic Equation of Fokker–Planck Type in the Limit of Grazing Collisions. Journal of Statistical Physics 104, 1173–1189 (2001). https://doi.org/10.1023/A:1010461929872
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DOI: https://doi.org/10.1023/A:1010461929872