This is not an answer to the specific question asked, but I cannot resist sharing this with you.
There is a beautiful result of Chernov and Dolgopyat, see http://www-users.math.umd.edu/~dmitry/galton11new.pdf , on the billiard in Galton's board in the presence of gravity. The Galton board is a well-known device indended to reproduce the binomial distribution (see, e.g., http://mathworld.wolfram.com/GaltonBoard.html).
Chernov and Dolgopyat studied the situation when the board is extended periodically and indefinitely. Collisions with the obstacles are assumed completely unelasticelastic, so that no loss of energy occurs. The result is that despite the fact that gravity is directed down, the ball or particle will eventually reach the same horizontal level as it is at initially. Moreover, it reaches a small neighborhood of its initial state infinitely many times. This happens for a.e. initial condition.