Consider an idealized classical particle confined to a two-dimensional surface that is frictionless. The particle's initial position on the surface is randomly selected, a nonzero velocity vector is randomly assigned to it, and the direction of the particle's movement changes only at the surface boundaries where perfectly elastic collisions occur (i.e. there is no information loss over time).


My question is - Does there exist such a bounded surface where the probability of the particle visiting any given position at some time 't', P(x,y,t), becomes equal to unity at infinite time? In other words, no matter where we initialize the particle, and no matter the velocity vector assigned to it, are there surfaces that will always be 'everywhere accessible'?


(Once again, I welcome any help asking this question in a more appropriate manner...)



Surfaces that are 'everywhere accessible' to a randomly positioned Newtonian particle with an arbitrary velocity vector
I'll interpret you question to be asking about whether the particle paths are "equidistributed" in the sense of dynamical systems. There is a large literature on this sort of thing, though usually instead of "particles" the authors talk about "billiards". While I don't know the answer to your question as stated, I do know that there are many examples where the paths become equidistributed for "generic" choices of positions and initial directions (in other words, the "bad" choices form a set of measure zero). 


Many examples and results of this form can be found in the wonderful survey "Rational billiards and flat structures" by Masur and Tabachnikov, which is available on Masur's [web page](http://math.uchicago.edu/~masur/).


EDIT : I forgot a nice reference! Serge Tabachnikov has written a very accessible book entitled "Geometry and Billiards" which is available on his webpage [here](http://www.math.psu.edu/tabachni/Books/books.html).


