Collisionless propagation in the Lorentz model: solution of the virtual BBGKY hierarchy

The kinetic theory description of collisionless propagation of a particle through a homogeneous static medium composed of randomly distributed hard spheres (Lorentz's model with overlapping) is derived. Both free motion and the motion accelerated by an external field are considered. The relevant virtual collision operator is given a clear physical interpretation. A rigorous solution to the corresponding virtual BBGKY hierarchy is found based on an exact reduction of the infinite hierarchy to a system of two coupled equations. In the case of two-dimensional cyclotron motion induced by a uniform magnetic field the finite, non-zero probability of unperturbed everlasting circling is derived from the hierarchy.