Surfaces that are 'everywhere accessible' to a randomly positioned Newtonian particle with an arbitrary velocity vector - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-22T11:21:18Z http://mathoverflow.net/feeds/question/3401 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/3401/surfaces-that-are-everywhere-accessible-to-a-randomly-positioned-newtonian-part Surfaces that are 'everywhere accessible' to a randomly positioned Newtonian particle with an arbitrary velocity vector Mensen 2009-10-30T05:07:03Z 2011-01-17T01:37:18Z <p>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).</p> <p>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'?</p> <p>(Once again, I welcome any help asking this question in a more appropriate manner...)</p> http://mathoverflow.net/questions/3401/surfaces-that-are-everywhere-accessible-to-a-randomly-positioned-newtonian-part/3403#3403 Answer by JoeG for Surfaces that are 'everywhere accessible' to a randomly positioned Newtonian particle with an arbitrary velocity vector JoeG 2009-10-30T05:18:32Z 2009-10-30T05:18:32Z <p>This is a classic and fundamental question solved using Ergodic Theory.</p> http://mathoverflow.net/questions/3401/surfaces-that-are-everywhere-accessible-to-a-randomly-positioned-newtonian-part/3404#3404 Answer by Andy Putman for Surfaces that are 'everywhere accessible' to a randomly positioned Newtonian particle with an arbitrary velocity vector Andy Putman 2009-10-30T05:19:32Z 2009-10-30T05:26:37Z <p>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). </p> <p>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 <a href="http://math.uchicago.edu/~masur/" rel="nofollow">web page</a>.</p> <p>EDIT : I forgot a nice reference! Serge Tabachnikov has written a very accessible book entitled "Geometry and Billiards" which is available on his webpage <a href="http://www.math.psu.edu/tabachni/Books/books.html" rel="nofollow">here</a>.</p> http://mathoverflow.net/questions/3401/surfaces-that-are-everywhere-accessible-to-a-randomly-positioned-newtonian-part/52276#52276 Answer by Joseph O'Rourke for Surfaces that are 'everywhere accessible' to a randomly positioned Newtonian particle with an arbitrary velocity vector Joseph O'Rourke 2011-01-17T00:42:51Z 2011-01-17T01:30:20Z <p>To supplement Andy's answer, there is a recent survey by Laura Demarco, "The conformal geometry of billiards," <a href="http://www.ams.org/journals/bull/2011-48-01/S0273-0979-2010-01322-7/home.html" rel="nofollow"><em>Bulletin AMS</em> 48(1), Jan 2011, pp. 33-52</a>. She defines a billiard table as <em>ergodically optimal</em> 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 <a href="http://journals.cambridge.org/action/displayAbstract;jsessionid=8FAA712C48489147DC7D334DAA03DB20.tomcat1?fromPage=online&amp;aid=2660296" rel="nofollow"><em>Veech's dichotomy</em></a>.</p> <p>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.</p> <p>There are examples that have billiard trajectories that are dense but not uniformly distributed.</p> http://mathoverflow.net/questions/3401/surfaces-that-are-everywhere-accessible-to-a-randomly-positioned-newtonian-part/52281#52281 Answer by Igor Rivin for Surfaces that are 'everywhere accessible' to a randomly positioned Newtonian particle with an arbitrary velocity vector Igor Rivin 2011-01-17T01:37:18Z 2011-01-17T01:37:18Z <p>I think the answers are not to the question asked (at least <em>as</em> 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 <em>every</em> 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.</p>