Timeline for probability of two confined randomly walking bodies overlapping
Current License: CC BY-SA 4.0
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| S Jul 10 at 18:01 | history | bounty ended | CommunityBot | ||
| S Jul 10 at 18:01 | history | notice removed | CommunityBot | ||
| Jul 6 at 21:37 | history | edited | RobPratt |
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| S Jul 2 at 16:44 | history | bounty started | gokudegrees | ||
| S Jul 2 at 16:44 | history | notice added | gokudegrees | Draw attention | |
| Jun 28 at 4:53 | history | edited | gokudegrees | CC BY-SA 4.0 |
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| Jun 28 at 4:52 | comment | added | gokudegrees | I will edit the question. By setting $\epsilon$ as the minimum resolvable distance between two disks, or a threshold distance which defines an overlap, this can be defined more exactly. | |
| Jun 28 at 4:42 | comment | added | Semiclassical | One issue with my formulation is that it's implicitly asking about whether "at least" $k$ disks are overlapping. One would need a sharper formulation to require exactly $k$ overlaps. | |
| Jun 28 at 4:40 | comment | added | gokudegrees | @Semiclassical I think this is a way to phrase it. I'll reiterate so that maybe i'm on the same page. Given $\epsilon > 0$, what is the probability there exists $k$ points $\epsilon$ apart. Then, $\epsilon$ defined a cut-off distance of the overlap, for example, half of the diameter. | |
| Jun 28 at 4:36 | comment | added | gokudegrees | @Aaron Good insight. I think, that you are right that this problem can be posed as just two uniform distributions. I do want to say that it can also be posed as a conditional probability. Ie, the walk gives the positional probability $p(r_n\lvert r_{n-1})$. Hence, if the particle step magnitude is not very large, the probability of overlap given that they are far away would be small, as opposed if they are very close together.I think treating it as two independent probability distributions is a good start. For $k$ overlapping disks, i mean that there are $N-k$ not-overlapping as you wrote. | |
| Jun 28 at 4:26 | comment | added | Semiclassical | This might be posed as: Suppose we allow $N$ points to 'move randomly' throughout a given domain. Given $\epsilon>0$, what is the probability that there exists a point within the domain which is within $\epsilon$ of $k$ of these points? | |
| Jun 28 at 4:25 | comment | added | Aaron | Nothing in the problem really depends on the specific dynamics of the random walk, only the distribution that the particles will have. Assuming that in the long term they have an approximately uniform distribution (is this true? I do not know), then we can turn our attention to the problem of determining the probability that 2 particles overlap. Though, I am a bit uncertain of your intention: when you say k particles are overlapping, do you want them all to overlap together, so there is some single point of intersection, or do you just mean that there are fewer than $N-k$ non-touching ones? | |
| Jun 28 at 3:53 | history | reopened |
Another User Harish Chandra Rajpoot Leucippus Ѕᴀᴀᴅ Anne Bauval |
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| S Jun 27 at 22:09 | review | Reopen votes | |||
| Jun 28 at 3:55 | |||||
| S Jun 27 at 22:09 | history | edited | gokudegrees | CC BY-SA 4.0 |
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| Jun 27 at 17:07 | history | closed |
Snoop Mike Earnest José Carlos Santos Harish Chandra Rajpoot Parcly Taxel |
Needs details or clarity | |
| Jun 24 at 22:55 | review | Close votes | |||
| Jun 27 at 6:07 | |||||
| Jun 24 at 21:49 | history | asked | gokudegrees | CC BY-SA 4.0 |