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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...)

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4 Answers

up vote 6 down vote accepted

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.

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.

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To supplement Andy's answer, there is a recent survey by Laura Demarco, "The conformal geometry of billiards," Bulletin AMS 48(1), Jan 2011, pp. 33-52. She defines a billiard table as ergodically optimal if, for each direction $\theta$, either every trajectory that avoids vertices is periodic, or every trajectory that avoids vertices is "uniformly distributed." It may be that your 'everywhere accessible' criterion is adequately captured by her definition of uniformly distributed. Ergodically optimal dynamics are also called Veech's dichotomy.

Any billiard table that can be tiled by squares is ergodically optimal; in particular, the square is (every rational $\theta$ is periodic, every irrational $\theta$ will lead to the particle "spending equal time in regions with equal area" [modulo avoiding vertices]). The regular $n$-gon is ergodically optimal.

There are examples that have billiard trajectories that are dense but not uniformly distributed.

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I think the answers are not to the question asked (at least as it is asked). The ergodicity of the geodesic flow (which, by the way holds for all negatively curved surfaces -- a fact surprising not mentioned in any of the above answers) does not mean that a fixed geodesic will hit every point on the surface eventually, but merely that it will become dense (well, more than that, but less than hitting every point). The OP asks for every point to be hit.

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@Igor: Yes, I see now. The definition of "uniformly distributed" that I quoted ensures that a measure of the trajectory converges weakly to normalized area--so "spending equal time in regions with equal area." But this is not the same as hitting every point. –  Joseph O'Rourke Jan 17 '11 at 2:00
    
Very true (and I noted in my answer that I wasn't precisely answering the question). –  Andy Putman Jan 17 '11 at 3:57
    
Examples of dense trajectories that forever miss some points in the domain are discussed in the MO question "Illuminating piecewise flat manifolds with geodesics": mathoverflow.net/questions/32797/… –  Joseph O'Rourke Jan 17 '11 at 12:10
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This is a classic and fundamental question solved using Ergodic Theory.

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