Mean Free Paths, Ion Drift Velocities, and the Poisson Distribution

In elementary kinetic theory of gases, the topic of free paths is usually limited to the simplest case of finding the probability of zero collisions in a travel distance x. This leads to a mean free path and it also discloses that the greatest probability of collision in a distance Δx occurs for x = 0. The most probable free path then does not coincide with the mean free path. This paper carries on the analysis to evaluation of the probability of exactly one collision in a distance x and still further to the probability of exactly n collisions in x. All of these probabilities, for which the number of collisions n is greater than zero, are bell-shaped curves with the most probable value of x coinciding with n times the mean free path. The results are applied to analyze the sharpness of the observed drift velocity of ions moving through a gas in an electric field.