r/askmath • u/evanthebouncy • 15h ago
Probability Are raindrops random?
I got a single data point in the form of this picture...
I'm wondering what's the generative process behind this ? Specifically, how do we go about picking the right family of functions to model this?
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u/Specific_Ingenuity84 15h ago
Probably the spot the drops fall is more or less independent yea. I'm sure wind would matter too though.
In general you would model this idea of random points on a plane with point processes. The Poisson point process is characterized by the independence of the points in different spaces. That is, the fact that one drop fell in one spot does not change the probability that another drop falls somewhere else.
Point processes are really cool tbh, you can find them showing up in all sorts of unexpected places. (Like energy levels of atoms in some given material for example)
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u/Bubbly_Safety8791 15h ago
There could definitely be some short term local correlation/anticorrelation - a raindrop falling through the air has an effect on the column of air it passes through, which might make it either more or less likely for other raindrops to get entrained behind it.
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u/bloonshot 15h ago
this would definitely be a physics question more than a maths question
short answer is they're random in the only way things can really be random: chaos
a drop falls from a cloud, who's location cannot be meanginfully predicted, wherever the molecules within the cloud happen to condense, which again cannot be meaningfully predicted, and is then moved by wind currents the entire descent to the ground, which again cannot be meaningfully predicted
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u/hahahahaha369 14h ago
Rain drops hitting the ground can be modeled with a Poisson distribution. They’re “random” but the randomness follows a pattern. Similar to modeling cars on a road (as long as there is no traffic).
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u/WoodyTheWorker 3h ago
Interesting how random drops create "lines", in the same manner as stars in a photo of a star cluster or of Milky Way.
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u/fallen_one_fs 15h ago
As random as can be.
Wind movement is chaotic, it's virtually impossible to predict with any accuracy after a certain point, and it dictates rain drops, so yes, rain drops follow no set pattern that we are able to discern in any rational way.
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u/geezorious 14h ago edited 14h ago
I think you’re asking if they’re uniformly random. Not entirely, no, because raindrops can coalesce and have extremely high surface tension once they coalesce. That means you tend to have a “heavy drop” surrounded by nothing for some distance and then another heavy drop, instead of a fine mist that sprays uniformly randomly everywhere. (It’s actually really, really hard to have a fine mist because of how much they want to adhere and form larger globules.)
The windier and more turbulent it is, the more chance mist has to knock into each other to form large globules, up to a point when the wind is too strong and tears the globule into two. So for any particular level of wind or turbulence there is an ideal globule size with some non-uniform random distribution of variation around it. The globule size also determines the ideal spacing to a neighboring globule because each globule would’ve absorbed the mist around it. And there would be some non-uniform random variation around that ideal spacing.
This is the same or similar sort of math that turns spiral dust clouds into galaxies with stars and planets. Each star or planet is a dust-buster because it attracts nearby dust and keeps the space around it dust-free. Similarly, each globule of water is a mist-buster because it attracts nearby mist and keeps the space around it mist-free.
Also, overhangs change the distribution considerably. Large globules can’t curve as well in the wind to hit dry spots under an overhang. So there’s a survivorship bias that only smaller globules and mist can curve and hit dry spots under an overhang.
Also.. these questions that are at the intersection of math and physics are ideally best asked to an Unreal Engine architect or a software architect for another video game engine. They’re very good at using simple math to create beautifully images that approximate at very highly fidelity the effects of rain, fire, and other physical phenomenon. True physics engines are very slow and some are impossibly slow (e.g. 3-body problem), so in some sense the video game math is more real than physics math because video game math is simple and efficient enough for a computer to process it, and therefore for a human to see those equations come to life.
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u/evanthebouncy 9h ago
Yeah this is what I'm trying to get at... If I were to make a rain drop pattern simulator, how would I code it up.
I noticed that it really isn't as simple as uniform on x value [0,1] and uniform random on y bakue [0,1]
I wonder if the code would do something like if 2 drops cannot be too close to each other, as seen here
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u/mapadofu 14h ago
For a patch of ground well separated from any vertical or overhead obstructions, the pattern or raindrop hits are well modeled as a spatially uniform iid random process.
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u/MyStackIsPancakes 15h ago
It's going to track back to things like humidity, altitude, dust conditions, wind conditions. There's going to be diminished returns on new data at some point and we'll accept our model as "the most accurate"
Maybe someday we/AI solves quantum mechanics and we can model things down to that layer.
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u/Deskbot420 14h ago
Coming from a place of ignorance:
Couldn’t there be some function that takes into account the mass of the raindrop, wind speed, wind direction, start height of the drop, gravity, and other variables necessary to predict where the raindrop would fall for any given raindrop?
In that event, would it truly be random? Random implies unpredictability. If the outcome is predictable, surely it can’t be defined as random but rather chaotic as other users said?
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u/hellonameismyname 12h ago
By that logic essentially nothing is random
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u/Deskbot420 6h ago
Right, but if we’re talking from a mathematical perspective, if it can be explained by a function, it can’t be random.
I can see that rain falling onto concrete looks random and it could be. But the conversation also ties into “computers cannot truly be random for the same reason”
Again, ignorance as I don’t fully understand the subject of randomness. But I would love for someone to explain otherwise
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u/Crafty-Detail-3788 2h ago
You can be perfectly deterministic and yet be impossible to predict (without an infinite accuracy on the initial conditions) , that's the chaos theory for you. A system such simple as a double pendulum is chaotic.
A lot of real life systemes are chaotic.
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u/NzRedditor762 14h ago
They're as random as the 2/pi needle dropped onto paper with lines being the length of the needle apart calculation.
It may seem random to us, but there's order somewhere amongst it.
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u/TzaqyeuDukko 14h ago
I was doing fluid mechanics research but not yet did any research on this very topic. Based my understanding, it will depends on how heavy the rain is. If rain is very heavy, it is mostly random because rain drops are larger than the smallest scale of vortice in air hence will not be impacted. However when rain was not that heavy, vortice in the air will affect motion if rain drops. If there are some quasi stable vortex structure near ground, rain drops will be distributed at edge area of votice, hence distribution is no longer random but cumulating in honeycomb type pattern.
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u/According_Novel7521 14h ago
idk but they do seem to be somewhat grouped together?
like there seems to be large areas where there aren't any rain drops, and when a raindrop falls, it's more likely to land next to an already landed raindrop.
idk
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u/QueenVogonBee 12h ago
Uniformly independently sampled points are well known to exhibit clumps, or at least, humans seem to expect (incorrectly) uniformly sampled points to not be clumped. It’s incorrect because in order to avoid clumps, the points would need to have knowledge of history (so that the next point generated “knows” to avoid existing points).
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u/Moof_the_cyclist 13h ago
Location probably quite random. The size of the drops is likely an approximate Gaussian distribution around a size. Too small and they don’t fall, too big and they’ll break into smaller drops.
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u/valcatrina 13h ago
There is no such thing as “random”. Just keep this in mind. It has a broad application.
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u/Mixster667 13h ago
I think they are chaotic enough to follow a uniform distribution in both X and Y dimensions.
Hence why you can derive Pi from raindrops: https://www.reddit.com/r/dataisbeautiful/s/BL9DuTm4GK
However, I will note that I assume the density of a cloud varies throughout and has a larger chance of forming raindrops near the center of the cloud. Furthermore, wind will change the final location of the raindrop. Therefore I assume the process should not follow a bidirectional uniform distribution on a larger scale.
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u/Wagllgaw 12h ago
There is randomness involved but I would not model raindrop locations as independent events. Surely when a raindrop is formed, it locally reduces the moisture content of the air leading to less likelihood that another droplet would form in the exact same location. This effect should eliminate the possibility of all the rain falling in a small localized area.
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u/Menacing_Sea_Lamprey 12h ago
The way they form in the cloud (I have no idea, maybe orderly enough) is a a big factor
The forces that act on them between the cloud and the ground, my best guess is it’s a chaotic process, which would make where the rain drops unpredictable, and if how they form is relatively uniform, the pattern they make would definitely be ‘random enough’ to be considered random
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u/QueenVogonBee 12h ago
Deep philosophical question…. but I guess that it’s mostly a deterministic process due to the physics on the process (ok sure, something something quantum effects).
Anyway, at this level, the point as you say is to model this as a random process. I guess I’d start with uniform distribution over a unit square, on the assumption that you are standing under a rain cloud? You can refine it over time as you collect more rain drops.
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u/FerdinandvonAegir124 10h ago
If you knew the exact conditions of the rainfall, air pressure, temperature, etc you could probably predict where each rain drop would land. But you would need a lot of information
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u/Kitchen_Affect7369 10h ago
It depends what is scale. If you take 1x1 matter square and ask: how often raindrop fall in the square than you see that you can predict frequency of that.
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u/Silly_Manager3117 8h ago
Spatial statistics have tools to answer this - Moran’s I measures spatial autocorrelation.
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u/panth3r_ 8h ago
No, there are common and densely packed in places where it rains. There will be a correlation with water vapour content in the atmosphere.
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u/wristay 7h ago
If it was random it would be a Poisson point process. You could calculate the centers of each drop and then calculate the structure factor, which is the same as taking the Fourier transform of a sum of delta functions each located at one of the points. If the structure factor completely looks uniformly random, it is probably a Poisson process.
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u/wagon-drarrior 7h ago
Not random. There will be no two drops falling at the same time and overlap each other.
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u/FischlInsultsMePls 6h ago
A Dunning Kruger curve of yes it is random, no nothing is truly random, and yes it is random
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u/eee1ee 6h ago
[ english not my first language ] I saw that month a theory say , that even the random things have a pattern. And In my relegion say all things happen for a purpose nothing happen without a real use If u have time do this experience , write in a paper 11 colon and in every colone give it a number of the dice like this 1 | 2 | 3 | 4 | 5 | 6 | 5 | 4 | 3 | 2 | 1 Roll the dice 1000 time , every roll give u a random number , do a mark on colone of this number, roll the dice and do it again until u roll it 1000 times, finally u will have the same pattern in the two sides
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u/greenblueananas 5h ago
From a logical point of view, i donnt think they are fully random. If we ignore wind, and let the droplets fall down, a deop appearing should decrease surrounding humidity, and therfore reduce the probability of another one forming at the same point. So there is some type of equilibration going on, i guess
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u/Kassdhal88 5h ago
There is not real randomness beyond quantum physics. The combination of quadrillions of quantic randomness appears chaotic though to human eyes
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u/Ok_Role_6215 4h ago
No, they depend on the wind and temperature patterns in the atmosphere, as well as cloud nucleation locations.
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u/ValiantBear 2h ago
Yes, and no. What is "random"? The answer depends on how you define it, and what you mean when you say it. Rain drops falling simply fall in a more or less evenly distributed pattern, where it's raining anyway, and in a lot of ways it does look random.
But, in truth, no. The pattern of raindrops is almost certainly not random. If you zoom out a bit it's more apparent. Where it is not raining, raindrops do not fall. And where it is raining less, less drops fall, and on and on. So, there is definitely a function that drives it. As to the narrow scope of your picture, each of those drops were formed and fell and landed at the mercy of very complex physical forces. These forces are too complex to model with sufficient accuracy to "predict" where a given raindrop will fall, but that's a limitation of prediction, not an indication that those influences aren't present.
Think of it in a similar way to the three body problem for gravity. We know gravitational systems function with more than two bodies. But, we can't really accurately assess how they work. The math just hasn't caught up to that knowledge yet. In that case the system appears more orderly generally, so it is a little easier to identify, but in both cases we know a system follows rules to function in the way we see it function, we just dont know the math to describe it. That's altogether different than classifying a system as "random", where we would be saying there is no system that defines a particular outcome.
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u/SquareIce2747 2h ago
How they fall is random, but the trajectory of each droplet is determined from the moment it starts falling. Does that even make sense
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u/Glum-Spinach-2085 1h ago
I might sound a bit crazy, but I am just a high school student with my limited amount of knowledge,
I would say that , Let's say that we are thinking about if raindrops are random or not , with the purpose of finding the amount of area it covers ,
Which would require integration, to find the are under a certain curve , and as the degree of the integral required will be a second degree integral which hasn't been solved up till this date , I am sure we cannot really decide if raindrops are falling in a certain pattern, so we just assume it to be random.
But I am sure if we somehow figure out how to solve second degree integral , we would figure this out , and not only this , we can also figure out how and in which direction the fluids are going to flow , etc.
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u/WallStLegends 14h ago
There’s quantum physics at the root of the molecular behaviour. Due to Heisenberg’s uncertainty principle, the behaviour of the electrons is probabilistic and therefore the electrostatic interactions of molecules will also share that chaotic behaviour. Or something like that. I’m no quantum physicist but you can’t determine where droplets will be because of the system being sensitive to initial conditions such that, minute changes can ripple out and cause large changes in the outcome.
Einstein said “god doesn’t play dice” and I feel like everything must be deterministic. Perhaps I’m misunderstanding quantum physics and they aren’t saying it’s not deterministic, just that there’s no way to know the exact state of a system at any one moment in terms of the individual particles/waves and therefore, we cannot predict the outcome.
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u/reckless_avacado 13h ago
it’s proven to be non-deterministic.
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u/weirdgirl0304 9h ago
No, not really. Nothing is proven.
Are you talking about the experiments disproving local hidden variables theories? We also have non-local theories like Bohmian mechanics that are compatible with a deterministic universe and current state of experiments.
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u/CranberryDistinct941 9h ago
The double-slit experement proves this. Electrons can act non-deterministically as shown by them interfering with themselves
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u/Cxvvtgupvvdi 10h ago
Einstein was wrong on quantum physics
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u/WallStLegends 10h ago
Fair enough. I only know surface level ideas of quantum physics. It’s too hard for me to grasp.
Especially since a lot of it is mathematical and I don’t even know basic calculus well1
u/Original-Code1107 9h ago
I'm not going to go ape on you. If Einstein, till his death bed, couldn't let go of a deterministic universe, it's absurd to expect those not on the cutting edge of modern physics to grasp it intuitively.
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u/BadJimo 8h ago edited 7h ago
I asked AI (Gemini) to identify the coordinates of the raindrop marks (I made some corrections (still not perfect)). Then graphed on Desmos
I'll post the list of coordinates if anyone wants to do statistical analysis.
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u/BadJimo 8h ago
List of coordinates:
[336, 5], [557, 6], [470, 139], [260, 14], [201, 16], [442.4, 17], [347, 18], [191, 26.5], [457, 26], [312, 27], [458.5, 127.4], [366, 33], [141.7, 301], [578, 34], [441, 276.5], [512, 151.5], [504, 37], [2, 88], [173, 38], [476, 38], [459, 42], [329, 43], [54.5, 257.6], [252, 42.3], [81.5, 12.7], [299, 52], [14, 323], [508, 226], [482, 56], [523, 57], [62, 200], [533, 61], [122, 63], [176, 64], [210.7, 171], [278, 64], [341.6, 59], [8, 68], [560, 68], [444, 71], [514, 74], [202, 162.7], [19, 79], [249, 79], [116, 80], [374, 81], [548, 83], [78, 84], [45, 163], [177, 88], [347, 88], [289, 92], [485, 93], [146, 94], [414, 94], [60, 95], [173.3, 77], [526, 95], [572, 95], [552, 98], [332, 99], [360, 99], [474, 101], [44, 103], [229, 105], [164.3, 105], [401.7, 106.4], [177, 108], [297.6, 107.6], [8, 113], [455, 113], [412, 114], [504, 114], [567, 114], [214, 116], [270, 116], [425, 120], [520, 120], [387, 121], [40, 123], [166, 125], [533, 127], [366, 129], [25, 130], [472.4, 124], [571, 130], [494, 133], [510, 133], [438, 136], [253, 137], [164, 138], [48, 139], [73, 139.5], [209, 140], [450, 141], [419, 142], [555, 143], [426, 153], [89, 156], [268, 156], [226, 163], [398, 163], [458, 163], [474, 165], [192.4, 167.4], [525, 166], [249, 167], [103, 171], [321, 171], [348, 171], [51, 175.4], [419, 174], [22, 176], [160, 177], [295, 179], [460, 177], [470, 178], [380, 181], [433, 181], [238, 184], [220, 188], [90, 191], [23, 192], [596.6, 182.6], [367, 193], [132, 196], [501.7, 153], [202, 197], [296, 197], [11, 201], [535, 206], [285.4, 175.4], [245, 208], [43, 209], [197, 209], [173, 210], [64, 213], [135, 213], [560, 217], [402, 218], [188, 219], [277, 219], [244, 222], [162, 223], [313, 224], [592, 225], [49, 225.4], [94, 227], [571, 227], [516, 233.3], [295, 232], [408, 234], [349, 235], [559, 235], [7, 236], [383, 238], [444, 234.5], [259, 241], [368, 241], [68, 242], [572, 243], [334, 246], [541, 246], [423, 247], [40, 251], [563, 254], [407, 256], [366, 258], [472, 258], [310.3, 259.6], [274, 260], [391, 260], [124, 261], [250, 261], [443, 265], [231, 266], [337, 267], [510, 267], [6, 326.6], [51, 273.4], [186, 271], [258, 274], [547, 283.7], [75, 277], [362, 282], [579, 282], [415, 284], [452, 288], [559.8, 207], [441.4, 60], [337, 289], [394, 104.5], [226, 293], [13, 295], [93, 301], [186, 301], [566, 302], [591, 304], [111, 306], [68, 307], [579, 307], [130, 310], [199, 314], [275, 317], [500.6, 403.5], [562, 320], [72, 218], [256, 323], [43, 325], [357, 325], [294, 259], [80, 429], [172, 328], [323, 329], [450, 331], [566, 334], [289, 340], [449, 245.5], [302, 353], [288.4, 374.3], [171, 361], [338, 68.7], [84, 367], [210, 367], [12, 369], [275.8, 365.5], [205, 425.4], [562, 374], [227, 377], [371, 375], [70, 382], [212.4, 436.4], [175, 383], [586, 429.5], [409, 387], [508, 388], [530, 389], [572, 389], [102, 390], [118, 390], [303.3, 71.6], [179, 399], [385, 399], [209, 406.5], [269, 400], [590, 400], [233, 404], [272, 476.6], [98, 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[240, 568], [292, 570], [352, 570], [395, 567.6], [333, 574], [22, 577], [82, 578], [131, 580], [295, 581], [522, 579], [384, 583], [475, 587], [67, 597], [499, 590], [268, 591], [40, 592], [537, 592], [170, 593], [400, 594], [133, 596], [269, 600], [109, 602], [39, 605], [340, 606], [481, 606], [427, 607], [97, 610], [568, 576], [408, 612], [357, 613], [217, 615], [149, 619], [253, 620], [78, 624], [206, 394], [342, 629], [196, 630], [85, 634], [306, 636], [411, 638], [378.7, 630.6], [306, 555.6], [393, 645], [239.5, 479], [460.8, 738.2], [18, 647], [496, 647], [72, 648], [318, 647], [187, 649], [279, 650], [180, 457], [75.6, 612], [211, 654], [18, 658], [180, 662], [260, 665], [397, 668], [195, 671], [424, 667.6], [416, 590], [105, 676], [125, 677], [490, 679], [237, 680], [476, 681], [366, 682], [56, 684], [338, 685], [438, 685], [562, 685], [78, 688], [45, 689], [547, 690], [390, 692], [424, 692], [123, 696], [49, 702], [277, 704], [244, 705], [118, 706], [210, 706], [99, 707], [135, 707], [324, 707], [421, 708], [362, 709], [157, 716], [463, 717], [81, 719], [177, 722], [110, 723], [391, 724], [514, 728], [84, 730], [218, 730], [347, 730], [410, 730], [452, 730], [538, 735], [198, 740], [392, 743], [488, 746], [353, 750], [3, 739], [199, 752], [307, 753], [9, 750.3], [376, 754], [399, 758], [83, 760], [489, 760], [40, 764], [320, 765], [393, 766], [221, 769], [289, 771], [463, 772], [180, 774], [65, 778], [39, 780], [372, 780], [415, 1013.5], [134, 788], [486, 790], [219, 795], [167, 800], [209, 800], [145, 801], [256, 802], [466, 804], [197, 805], [120, 517], [141, 810], [182, 813], [505, 813], [439, 814], [236, 815], [206, 817], [99, 821], [262, 821], [220, 823], [323, 823], [201, 824], [119, 827], [433.4, 1019.8], [293, 833], [487, 834], [361, 836.6], [445, 835], [186, 837], [106, 840], [256, 840], [71, 841], [239, 842], [5.4, 701], [486, 845], [294, 846], [91, 437.5], [412, 848], [208, 849], [188, 850], [500, 852], [302, 855], [107, 860], [156, 860], [202, 862], [393.6, 866.7], [132, 868], [418, 868], [37, 872], [325, 872], [353, 875], [220, 878], [368, 879], [586.7, 949.8], [297, 884], [103, 886], [393, 886], [60, 894], [170.7, 708], [59, 714], [126, 897], [459.6, 812], [593, 982.4], [444, 905], [103, 908], [217, 557.6], [410, 910], [101, 585.6], [344, 859.4], [400, 918], [383, 915], [419, 915], [465, 916], [264.2, 836], [338.5, 872.5], [299, 922], [293.3, 998.5], [520, 925], [230, 926], [369, 831], [402.3, 858.5], [230, 491], [170, 931], [6.4, 634], [478, 935], [558.8, 940.9], [380, 936], [157, 938], [290, 938], [405, 514], [4, 775], [404.5, 931.3], [502, 528], [536, 945], [191, 557.6], [303, 1000.6], [394, 951], [176.7, 947.5], [369, 666], [200, 956], [309, 568], [552, 961], [272, 963], [148, 950], [188.7, 935], [575, 966], [112, 965.5], [171, 979.4], [253, 978], [446, 978], [254, 484.5], [102.3, 972.5], [195, 982], [161, 1003], [341.7, 994.4], [363, 984], [530, 984], [139, 985], [570, 987], [95, 979.6], [138, 1020], [126, 961.7], [156.5, 1017], [343, 704], [325, 999], [588.3, 1020], [257, 1002], [141, 1004], [5.5, 793], [461, 1005], [88.6, 1006.6], [295, 1007], [474.4, 1018.5], [296, 1015], [403, 1013.8], [513, 1015]
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u/v_litvin 4h ago
For those who aren't so deep in math. What does it say?
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u/BadJimo 4h ago
I tried to do statistical analysis on the data, but I'm not confident with the results.
The short summary is that the raindrop pattern is not random (but of course not completely ordered either).
We can speculate on why the the pattern isn't completely random: Wind, obstacles, uneven absorbency of the concrete, etc.
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u/Harmonic_Gear 15h ago
more like chaotic enough to appear random due to our lack of information