Timeline for The lonely molecule
Current License: CC BY-SA 3.0
12 events
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| Apr 13, 2017 at 12:58 | history | edited | CommunityBot |
replaced http://mathoverflow.net/ with https://mathoverflow.net/
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| Jul 25, 2014 at 20:52 | comment | added | Joseph O'Rourke | @ChristianRemling: I believe commenters were responding to the initial conditions exactly as I stipulated, rather than on the "interesting situation" that you outline. In other words, my question was flawed in a sense, and my addendum reflects the answer to my flawed question. Feel free to respond to the more interesting question! | |
| Jul 25, 2014 at 18:47 | comment | added | Christian Remling | I'm a bit confused by your addendum, as well as by Evan's comment. Of course everything is clear in the ergodic case; the same remark applies to the lonely (or social) runners. The interesting situation that we must focus on is when we have as many rational relations among the speeds as possible (= periodicity for the runners). | |
| Jul 25, 2014 at 10:43 | history | edited | Joseph O'Rourke | CC BY-SA 3.0 |
Answered in the comments; summary.
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| Jul 25, 2014 at 9:38 | comment | added | Joseph O'Rourke | Thanks, everyone, for the cogent comments. It is now clear that the question needs reformulation along the lines Will suggests to be as interesting as the lonely runner situation. | |
| Jul 25, 2014 at 2:45 | comment | added | Will Sawin | Precisely which non degeneracy condition you choose is critical. In the lonely runner conjecture, the velocities are required to be distinct, but not linearly independent over $\mathbb Q$. I would guess the right approach is to require the velocities be pairwise distinct in each coordinate axis (or just one?). One might also consider what notion of distance is appropriate - $L^1$ norm? $L^2$ norm?. | |
| Jul 25, 2014 at 2:25 | comment | added | Evan Jenkins | @JosephO'Rourke: Consider your $n$ particles in a $d$-dimensional cube instead as a single particle in the $nd$-dimensional unit torus (an unfolding of $2^{nd}$ cubes). Provided the $nd$ coordinates of the velocity vector are linearly independent over $\mathbb{Q}$, it follows that the $\mathbb{Q}$-span, and hence the $\mathbb{Z}$-span, of this vector is dense in the torus. This implies that any nonempty open condition on particle configurations will be obtained. | |
| Jul 25, 2014 at 1:16 | comment | added | Anthony Quas | If they're randomly placed, then they will almost surely cluster somewhere sooner or later. It's easiest to work on the torus, but completely equivalent because you can unfold $2^d$ copies of the cube to make 1 copy of the torus. | |
| Jul 25, 2014 at 0:53 | comment | added | Joseph O'Rourke | @ChristianRemling: Excellent point! Yes, let them pass through one another, like ghosts. | |
| Jul 25, 2014 at 0:48 | comment | added | Christian Remling | No such condition (on the angles) can prevent collisions: we can still shift parallel to the coordinate axes. However, just like the runners, we can just let the molecules ignore each other when they do collide. | |
| Jul 25, 2014 at 0:46 | comment | added | The Masked Avenger | Why put up walls? Have them run around in a 3-torus. You might get quicker answers that way. | |
| Jul 25, 2014 at 0:32 | history | asked | Joseph O'Rourke | CC BY-SA 3.0 |