On Trapped Trajectories in Brownian Motion

The Brownian motion of an ion in the attractive field of an infinite line charge is investigated with reference to the existence of trapped trajectories, i.e., trajectories which never depart from the neighborhood of the center of attraction. It is shown that there is a discontinuity in the nature of the Brownian motion as a function of the line charge density, marked by the appearance of trapped trajectories when the line charge density becomes greater than a critical value.