On the Wang-Uhlenbeck problem in discrete velocity space

The arguably simplest model for dynamics in phase space is the one where the velocity can jump between only two discrete values, +/- v, with rate constant k. For this model, which is the continuous-space version of a persistent random walk, analytic expressions are found for the first passage lime distributions to the origin. Since the evolution equation of this model can be regarded as the two-state finite-difference approximation in velocity space of the Kramers-Klein equation, this work constitutes a solution of the simplest version of the Wang-Uhlenbeck problem. Formal solution (in Laplace space) of generalizations where the velocity can assume an arbitrary number of discrete states that mimic the Maxwell distribution is also provided.