Timeline for Random N-body problem
Current License: CC BY-SA 3.0
18 events
| when toggle format | what | by | license | comment | |
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| Jun 20, 2017 at 10:37 | vote | accept | Joseph O'Rourke | ||
| Jun 20, 2017 at 10:37 | history | edited | Joseph O'Rourke |
As per Ben C.'s answer, it appears to be open for N > 3.
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| Jun 18, 2017 at 21:17 | answer | added | user21349 | timeline score: 6 | |
| Jun 17, 2017 at 14:58 | answer | added | Geoffrey Irving | timeline score: 1 | |
| Jun 17, 2017 at 12:18 | comment | added | Joseph O'Rourke | @ChristianRemling: That is indeed a better fix; thanks. And user1504 also makes a good point. And I should have specified $N>2$. So my question is flawed in several ways. Rather than continuing to change it, I will leave it as-is, and hope for a clear resolution of some reasonable version. | |
| Jun 17, 2017 at 6:45 | answer | added | Alexandre Eremenko | timeline score: 3 | |
| Jun 17, 2017 at 2:01 | comment | added | fedja | @ChristianRemling Not at all even if $N=3$. The energy bounds alone do not preclude the possibility that two particles will go into a very low energy configuration and the third one will escape. | |
| Jun 17, 2017 at 1:13 | comment | added | Christian Remling | Maybe a better fix would be to ask if the $k$ particles stay in $B_R(C(t))$, where $C(t)=a+bt$ is the center of mass. In the current version, it's not clear to me if there is a natural interpretation of picking the velocities "at random." | |
| Jun 17, 2017 at 1:07 | comment | added | user1504 | Also, fwiw, I wonder if the 'galaxy formation' question astronomers consider may be slightly different. Planets and stars have finite size and they tend to merge when they collide. (Or more generally scatter into chunks of unequal size.) It's not totally clear to me if the asymptotics are the same. | |
| Jun 17, 2017 at 1:04 | comment | added | user1504 | You probably also want to fix the total initial energy $E$ (and maybe also the angular momentum $L$). These are conserved quantities. The higher the energy E is relative to the energy scale set by the radius $R$, the more likely it is that the system 'explodes' out of $R$. | |
| Jun 16, 2017 at 23:18 | comment | added | Joseph O'Rourke | @RobertIsrael: Now incorporated into the question. Thanks! | |
| Jun 16, 2017 at 23:17 | history | edited | Joseph O'Rourke | CC BY-SA 3.0 |
Robert Israel's comment incorporated.
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| Jun 16, 2017 at 23:08 | comment | added | Joseph O'Rourke | @RobertIsrael: Brilliant! But suppose I had assumed that the velocity vector sum was exactly zero? (Which I clearly should have...) | |
| Jun 16, 2017 at 23:07 | comment | added | Robert Israel | If you're talking about a fixed disk of radius $R$, the probability should be $0$. The centre of mass undergoes uniform motion, with velocity the initial average of the velocity vectors, and that average will almost surely be nonzero. This isn't quite conclusive, e.g. it is possible that in the limit various clusters of particles will move away with certain velocities while one cluster has average velocity $0$, but it should be intuitively clear that an average cluster velocity of exactly $0$ has probability $0$. | |
| Jun 16, 2017 at 23:07 | history | edited | Joseph O'Rourke | CC BY-SA 3.0 |
n => N, just because N-body capitalizes. And cite for simulations.
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| Jun 16, 2017 at 22:52 | comment | added | Gerhard Paseman | Indeed, for d=3, I submit that your post is evidence that for some large N and some large k less than N, the probability is greater than zero. Gerhard "Witnessing The Meaning Of Life" Paseman, 2017.06.16. | |
| Jun 16, 2017 at 22:47 | comment | added | Gerhard Paseman | I bet cosmologists thought about this problem in considering origins of the universe. Gerhard "Into The Universality Of It" Paseman, 2017.06.16. | |
| Jun 16, 2017 at 21:28 | history | asked | Joseph O'Rourke | CC BY-SA 3.0 |