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We consider the motion of a classical particle in a random isotropic potential arising from uniformly distributed scatterers in two dimensions. We prove that in the weak coupling limit the velocity process of the particle converges in distribution to Brownian motion on a surface of constant speed, i.e. on the circle. The resulting equation for the probability density of the particle is related to the Landau equation in plasmas.

Asymptotic motion of a classical particle in a random potential in two dimensions: Landau model