ON THE WANG-UHLENBECK PROBLEM IN DISCRETE VELOCITY SPACE

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