Timeline for A random process with conserved momentum: 'particle decay'?
Current License: CC BY-SA 4.0
10 events
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| Sep 15, 2021 at 17:16 | comment | added | Willie Wong | Aren't there infinitely many rhombi with unit side length? The solution when $N = 3$ is not unique, even if you quotient out by the symmetry swapping the particles. (Solution space should be $N-2$ dimensional, if I didn't make a mistake.) // I'd love to see an animation... | |
| Sep 15, 2021 at 16:49 | comment | added | Leo Moos | @Sinusx You're right, thanks for pointing this out; I corrected the question. | |
| Sep 15, 2021 at 16:49 | history | edited | Leo Moos | CC BY-SA 4.0 |
corrected minus sign error
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| Sep 15, 2021 at 15:24 | comment | added | Viktor B | Wouldn't the 'conservation of momentum' equivalent be $v_1 = \sum_{i} v_{1i} $? For instance if $N=1$, then by your equation the new particle starts moving in the opposite direction. | |
| Sep 13, 2021 at 14:36 | comment | added | Leo Moos | @CarloBeenakker They'd be picked uniformly at random from among the intersection $\{ \sum x_i = -v_1 \} \cap (\mathbf{S}^1)^N$. | |
| Sep 13, 2021 at 14:22 | comment | added | Carlo Beenakker | I'm confused; how would you implement the constraint $x_1+\cdots +x_N=-v_1$ if you pick them uniformly at random from the unit circle? Supposse the first $N-1$ you pick happen to be nearly aligned, then the velocity of the $N$-th particle might have to be much larger than unity to achieve the desired sum. | |
| Sep 13, 2021 at 13:47 | history | edited | Leo Moos | CC BY-SA 4.0 |
added paragraph to clarify and explain distribution of momenta
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| Sep 13, 2021 at 13:40 | comment | added | Leo Moos | @CarloBeenakker Good point, I forgot to mention that the particles would move at unit speed, meaning $v_{11},\dots,v_{1N} \in \mathbf{S}^1$. That being understood, the momenta would be picked from $\{ (x_1,\dots,x_N) \in \mathbf{S}^{1} \times \cdots \times \mathbf{S}^1 \mid x_1 + \cdots + x_N = - v_1 \}$. As the product $(\mathbf{S}^1)^N$ is compact, I guess they could be picked uniformly at random; would that make sense to you? | |
| Sep 13, 2021 at 13:32 | comment | added | Carlo Beenakker | with what distribution would you choose the decay momenta? they are obviously not i.i.d. ... | |
| Sep 13, 2021 at 13:18 | history | asked | Leo Moos | CC BY-SA 4.0 |