in the normal operation, as it was so often told to me, the ions speed into the cathode space having overshot the physical cathode, and some will collide, some others will pass through the center of the cathode space and reverse course and pass through the center again ... and again, and after an indeterminable number of repetitions, collide with something and fuse, of course there are many other possibilities, such as pick up an electron and cease to be ions.

the mathematician who is named Poisson, Siméon Denis Poisson to be exact, was an early student of probability.  So I visited the site below to read up on his work.