Why Do Exact Equations Become Unpredictable?
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- čas přidán 29. 12. 2023
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The space simulations were done in Universe Sandbox:
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Paper showing the probability of Mercury, Venus and Mars hitting the earth:
sci-hub.se/www.nature...
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Chaos: deterministic but sensitive to initial conditions. Other examples: double-pendulum, circular restricted three-body problem (CR3BP), satellite attitude dynamics, world generation in Minecraft using an input 'seed'.
chaos really is a primitive word for entropy.
I wouldn't classify world generation from an input seed as chaotic.
@@jwonz2054 It is, any change in the seed creates completely different results, but the equation is deterministic and the same seed always gives the same result.
@@metasamsara Entropy?
entropy and syntropy are more scientific words than "chaos" and "order". There isn't one start and one end, it's a never ending balance of energy polarity into self-stabilization, like magnets and molecules along with things like ionization and light polarity forming colors, etc. that's why time being linear is a dumb concept that needs disproving. no, planets rotating around the sun experienced by humans doesn't define the tickrate of entropy and syntropy everywhere in the universe. Just like two planets can have different gravity pull depending on their mass, entire nebulae could very well experience time totally differently at the same time we do, because it is all relative and depending on entropic charge. Light already exudes properties that break the concept of time and even go back through time. There is no proof that entire nebulae couldn't experience time totally backwards whilst we experience it forward for instance, since after all, it's only a specific value of entropy or syntropy on a balance that defines this chemical transformation we hallucinate as being "time". For time to exist, there needs to be a past and a future, but those are experiences of the ego, in reality, there is only the present ever transformation of entropy/syntropy and it isn't set in stone, only existing at scales much much much much much larger than we can even conceive of, let alone dare to measure. in fact there is no proof that scale isn't another infinite dimension. we pretend we know the smallest unit and the largest items in reality but it is only a false pretense. solar systems work through the same exact laws as molecules do, only with the same forces manifesting differently at different scales.
For those interested in this topic, I highly recommend "Chaos Making A New Science" by James Gleich. Its a very easy book to read but still really dives into chaos theory by way of talking through the history of it, explaining each discovery every step of the way with very eye-opening illustrations.
Thank you for the suggestion!
On a whim I checked it out and was completely sucked in by the prologue. I then proceeded to get a few other of James Glieck's books (especially looking forward to reading "The Information: A history,A theory, A flood")
Thank you!
bought it! Really want to know more about this
I was reading this book and this video came out. Perfect timing!
Ye I love this book! I always had an obsession with Chaos Theory and this book satisfied my thirst 😜
A minor note: That 2009 paper was using a DIFFERENT position of Mercury's orbit (3.8cm TIMES k ∈ {-100, 100}) for EACH simulation. I feel like that wasn't stated as clearly as it should have been. That k factor changes the orbit of Mercury with each simulation well within its known deviation of what we've measured it to be. This is why the simulations came to such huge disagreement even though they use deterministic equations. The minor influences in the rest of the overall system begin to add up and produce this "chaos".
And yes, the paper proposed that the probability of a large increase of Mercury's eccentricity (due to resonance with Jupiter) which leads to these potential collisions is about 1%.
@@Aviv1201 o
@@chitlitlahc
@@chitlitlahr
Whole video i was thinking about that simulation with a thought that he said something wrong because different results with only 1 prameter changed only once contradicted what he said afterwards
I was thinking the same. The way he phrased it gave the impression that the equation was completely stable at first, but after a single one change in the input parameters, the same equation started to give different answers each time running it without any further changes to the parameters.
It's good that you post this stuff, because it illustrates that math based on hard boundary numerical input can never really describe reality on the very large or very small scale. This is why a lot of my career in signal processing has been involved with injecting a degree of statistical probability density functions into the digital signal processing systems we were making. In other words - we are 'blurring' the probability of the terms and input to access the 'real life' results of our processes to get a real world spread of results. We should never forget that for the real world a change from 0.000000001 to 0.000000002 is still an infinite boundary. This stuff is built into the audio signal processing in products I have made since the 1990s :-)
Yeah, I learned this in grad school. In the feynmann lectures he goes over a very interesting concept in the Quantum Mechanics part where he says that Quantum Mechanics having uncertainty is no different than if we had a completely "pure classical physics" mindset in the sense that we have no infinitely precise measuring device. Therefore even the most accurate device will still have inaccuracies, and those inaccuracies compound (very very quickly) as time goes on as each force interaction takes place.. Again, completely ignoring quantum physics, the universe could never have been deterministic with any physical measuring device we could achieve in reality. Its a simple notion, but very eye opening.
"the universe could never have been deterministic with any physical measuring device we could achieve in reality"
Life happens, is happening, happened, will happen even if no one is/was measuring it "with any physical measuring device" - so IT IS deterministic whether the equations (classical, quantum, or otherwise) say so or not.
@@bluceree7312 you are reinventing "Hidden variable theories" if I am not mistaken. And if I recall correctly, some classes of those have been proven false.
Your precision in your equations is the problem. You need to use smaller decimals (billions of zero's) to get similar results for such large durations as billions of years. The number does exist on all objects. We just can't measure to the level of precision of a number needed, that is infinitely large.... but accurate to exact precision.
@@doyouwanttogivemelekiss3097
you recall it wrong
@@IamUzyf c.f. kochen-specker theorem.
There's a fantastic book on this topic called "Complexity: The Emerging Science at the Edge of Order and Chaos" by Mitchell Waldrop. Goes into the science and study of Chaos, and why it died out as a field of study. Really interesting and poignant discussion of these topics, as well as the challenges that prevent us from understanding it better.
entropy and syntropy are more scientific than "chaos" and "order"
@@metasamsara Yeah, but it's not a science textbook. It's a expository description of the science and the story around it. Part of why it's so good is because it simplifies a post-doc level field into something that can be read and appreciated by an undergrad.
@@nathanrice7352 I understand but a lot of people misunderstand because of this "simplifying". They think order is good, chaos is bad, instead of understanding it is a layer of balancing of charges into perfect stabilizing which creates reality. There isn't one start and one end, only multiple things affecting the equilibrium both ways, just like other sources of energy polarization. You wouldn't say a magnet is either good or bad, you understand it's positive or negative. It's not complicated to simplify in more sensible ways.
@@metasamsara i don't see him saying its bad
Something you missed: to predict exactly where the planets in the solar system will be in some amount of time, you not only need to have exact inputs, but also you need to take in account everything that has mass in the universe, since everything that has mass has gravity, and the gravity of an object is never 0 no matter where you are, everything that has mass has an effect on the mouvement of planets. It might not make a very long difference in the short term, but if you wanna check for 5 billion years from now, it probably makes some difference
Fantastic! This was a great listen and I love the way you lay out the information. As far as 'coming to the end of science' I agree we have a long way to go and likely always will.
Great video! Been watching action lab for a while but this one was particularly fascinating. Keep up the great work👍
The most common example of this is The Butterfly Effect which you didn’t mention. As a math and science teacher I REALLY appreciate that because it takes many people 2 or more distinct exposures to an idea before they begin to understand it. I’ve discovered that a combination of stories, pictures, animations, numerical models, problem solving, equations, tables, maps, plots, charts, heat maps, topological graphs, etc. can enable understanding when students might be otherwise lost. Chaos and complexity are very deep and broad topics that connect with almost everything in STEAM, so showing people some of the various ways that they work is ideal. Well done!
Weather forecast is another great example
@@nomars4827 That is so true! In my algebra II, calculus and Earth science classrooms we discuss short-term meteorological observations vs. long-term climatological models and challenges with both modeling and setting policy (extrapolating the latter from the former and attributing any changes to human activities) in view of nature’s preexisting complexity and chaos. (I worked with climatologists in the late 90s who were all deeply troubled by the increasing politicization of their field.)
@@Aviv1201 흠
Sounds like you've been a good teacher :)
Btw what's the A in STEAM ?
@@YounesLayachi Art, they added art.
The content, as usual, is fascinating. I am a staunch Telsa fan, so naturally, I love the shirt!
Asteroid: * flys by mercury *
Science: "Were all gonna die!"
Great video topic. I think understanding chaos is one of the most important topics in science.
theres this really good triology about this named "Three Body"!
Excellent presentation of the concept again!
This was a wonderful and simple explanation. Thank you 😊
I highly recommend 'Chaos: Making a New Science', a lovely book by James Gleick. It's extraordinarily well written, and explains the topic with perfect clarity.
It gets a bit preachy near the end, but I second the motion.
We will never come to the end of science! There will always be something out there to discover! Both haunting and thrilling at the same time!
One of the best science channels on the Tube.
Great flic! Thank you. You are becoming more and more professional
Congratulations! Keep up the good work.
Amazing. Could watch your videos all day.
brilliant video! chaos explained!!!!
This was SOOO COOL! Thanks for video!
Happy New Year Kevin and Backardigans
Merry Xmas and happy new year 🙏
Seems like they make it up as they go. Thanks for the stats!
I looove this concept of chaotic and would love for you to cover it further showing fractals and the Mandelbrot set
Correct me if I'm wrong, but even if we knew the exact positions and velocities, we'd still not be sure about the result because the equations do not have a closed solution so the process of solving these equations is by itself an approximation.
As far as I understand it, if you have infinite accuracy in a world of spherical featherless chickens in a vacuum, and you assume the equation is a completely accurate model of reality (like how newtonian gravity is accurate up to a point compared to general relativity) then you would have the actual result. However, it is proposed that even the universe does not have infinite resolution (see planck length) which would then indeed mean that the equations do not *always* have a closed solution
With "not always" I mean that the chaotic equations often have stable, instable, and fractally instable regions, based on starting conditions. It is only in the fractally instable regions that the solution can not be derived. For more information on the different regions, search for "magnetic pendulum chaos theory" and you will see what I mean.
Some of the equations do have an exact, analytic form and yet are still sensitive to initial conditions, i.e. chaotic systems. The two-body orbital problem is one such case.
3:03 You said something technically true-ish but confusing to the noobs. There's no equation that will tell you the positions of the planets. The equations just describe instantaneous motion of the planets, which lets you simulate their motion step-by-step over a long period of time to determine their position at some later time. I don't mean to nitpick, but it's an important distinction to make for people new to the topic, because a small change to the inputs of smooth equations like the ones you showed produces only a small change to the outputs.
that's what i found weird. how come the study included a tiny difference in spatial location, when the timestep of the simulation is probably waaaaaaaaay less accurate and way more important
The equations that govern planetary motion are unstable in a Lyapunov-sense. Small perturbations in the 6-dimensional physical state (3 position, 3 velocity components) will cause the planets to have wildly different final states, hence chaotic motion. Even the two-body orbital problem is unstable despite a very nice polar analytic equation that describes the relative motion. In this case (2BP) the orbit itself is generally stable (orbital stability), but the relative location of the planets in that orbit is not (Lyapunov stability).
Tremendous video. Thank you.
Thanks for explicitly mentioning that bit about quantum mechanics at the end!
So chaos isn't random. it is, in fact so precise that we aren't good enough to use it.
Happy new year
A couple of years ago I built a scene in Blender that was used to simulate throwing RPG dice. There were ramps and walls, etc. I changed something that shouldn’t have had any effect on the bounces. It was either dice colors and transparency or lighting positions - basically only a visual edit. The bounces changed pretty dramatically. I was wondering why that affected the physics. No idea. I just tell myself it was chaos theory at play. Had to undo those edits.
Hey - that was fun. So well explained. Learned something I hope to chat with my kids about.
I would be very happy if I got him as my teacher. Definitely appreciate your hard-work sir! ❤❤
You do! Just watch everything on his channel and then do extra research to understand more.
The chap’s PhD is chemical engineering, and it requires loads of both physics and maths, so he is great in all of those subjects.
Thankyou. It’s amazing that the meaning of chaos doesn’t ring the bell with many persons Relating as to what it means in this stated perspective
Somehow the messaging of "deterministic with no randomness" and "sensitive to inputs" never quite clicked before this video, but it was super clear here. Very clear and helpful explanations!
You are awesome, thanks!
My exams are coming ..... PLEASE MAKE IT HAPPEN
Same
WTF
Go and fucking study
You are worse than hitler
You won't be any more for the exams .😮
Yup, you made me understand, what chaotic equations are.❤
I don't think science will ever end. Rather it'll evolve and keep evolving as time goes by and changes happen.
I love this equation! The Logistic Map!
That shirt is both awesome and hilarious! Love it!
BUTTERFLY EFFECT 🦋
Always nice to have course correction thrusters.
I wonder if this can be used in reverse for precise measurements? Since tiny differences in initial values eventually lead to large discrepancies at some future point, measuring the state of a system at a future point will provide a hint as to what initial value was most correct.
IMHO, this video is the best you have ever produced. It's mind blowing that our solar system even exists, and has for billions of years, since as you said, just a .3 millimeter initial measurement error would impact the entire solar system over time.
The deterministic models work fine for short time spans, but if man were ever to navigate amongst the stars, he had better figure out how to get to other places and arrive where he thought he was going. Math is great, but as you also pointed out, this is more a philosophical issue than a mathematical one.
This is a really convoluted way to explain rounding error.
Mind-blowing literally
Hope your visit to Holden was nice!
Friends or acquaintances curious about Nichiren Buddhism often ask what Nam-myoho-renge-kyo means. This is a very important and difficult question
"We have almost discovered how everything works" Didnt i just watch a video somewhere that stated that 2/3 of everything that we know is out there due to gravity is unseen or unknown by us? Dark matter or energy? Science has come a long way but it has a long way to go.
I think the point is that we've figured out a lot about what we can measure. It could be that dark matter and dark energy are fairly simple, once understood, and that would add just two more "facts" to the current huge body of knowledge. However, we know very little about dark matter, and we're not even 100% sure it exists (there are competing hypotheses). And dark energy is similar. They may be one phenomena each, or they may be entire families of things that add up to the effects we see. No one knows yet.
@@yeroca Doesn't not knowing 2/3 of everything means we have discovered 1/3 of how everything works? Doesn't the math tell us that we are a far cry from having almost discovered how everything works? I feel competent enough to respond because I have enough fingers to do the math. Haha I can even do a fraction with one of my fingers.
@@Estwing22 At this point there's no way to calculate how much is known because we don't know how much we don't know. We might know 0.00001%, 5%, or 40%. At least, though, we can describe mathematically what's around us pretty well. We don't exactly know "why" we are able to describe it mathematically, except that math does do an astoundingly good job of making predictions. Take the Higgs particle for example.. it was predicted from the math about 60 years before it was found in the LHC experiments. Likewise, antimatter was predicted decades before experiments proved its existence. It's a very deep area of thought, and I don't know much about it other than the "laymen level" articles I've read over the years.
Sometimes I feel like this is the exact explanation for what happens when I'm trying to make something to a precise size on my old lathe with my rudimentary experience and knowledge. I start to swear the universe is very unpredictable.
Thanks for the information an video
When you are dealing with probabilities like this in chaotic equations and systems to come up with "accurate" predictions you change 1 variable several times depending on how far in the future you want to "look" you use more or less variations. This will give you a range of outcomes in percent (im skipping a ton here is a bunch of things to do to get that percent chance) So when you see 10 to 25% chance listed in a probability outcome that is because of variations used to generate the probability you often see this in surgical or medical outcome probabilities most often times the doctor will say you have a 10% chance of survival with out the surgery or a 50% chance with. But what medical researchers have actually done is taken the numerous percent chance of survival and averaged them out to come up with a single number. Depending on how many variations were used (mostly based on previous historic outcomes from the surgery) the accuracy of said percent chance will be quite high.
So called outliers in outcomes and results in probability studies really are not outliers at all but well with in the probability only we do not have enough variations over time to show this.
Unlimited attempts
Man that graphic of the tape measure and the planets was cool asf!!!
I'm ready for it 🤞
Frankly, if the problem is that some systems are so unstable that they are modeled to wildly different outcomes in the absence of absolutely perfect knowledge - that doesn't sound like something ANY possible advance in science could do anything about, ever.
To be more precise : you can't predict what a chaotic system will do after a specific amount of time for a given accuracy and precision of the initial conditions guess. However, if you can steer the system accurately enough within that time span, you can effectively tame the chaos.
One spectacular demo of that process is the control of a double or triple inverted pendulum. The more precise the initial guess is, the longer the window will be.
Absolutely perfect precision is impossible--but the greater the precision, the further into the future we can go and still be confident about what will happen.
While calculating a solution approximation to these types of problems will always be a losing battle there are ways to make it a bit better. These types of simulations typically involve the time domain being broken into discrete steps. Smaller steps usually reduces the rate at which the approximation deviates from the "true solution" but comes at the cost of computational time. By using numerical software with error control you can attempt to minimize the error in the approximation without incurring unnecessary computational cost.
On the contrary, with science you can:
- Get better mathematical models of reality.
- Get better determination of the constant in those mathematical models (such as the universal gravitational constant).
- Improve the measurement systems to improve the accuracy of the determination of the initial conditions.
- Improve the statistical models to estimates the probabilities for a given measurement uncertainty of the initial conditions.
- Help advance in computation power to be able to do more precise computations of the differential equations (that don't have exact solution, so they need to be solved with numerical methods which, as explained in the previous comment, are based on calculating it on discrete steps of time, but the shorter these steps the more precision).
That will not resolve the chaotic nature of the phenomena, but it can take you from "I have no clue if that comet will hit the Earth in the next 100 years" and the Earth will colide in the next 10,000 years" to "I am certain that it will not hit the Earth in the next 10,000 years, and 99.9% sure that it will not in the next 1,000,000 years either".
That's not perfect but is not nothing either.
@@RvBAfras The smaller steps only help with non-chaotic systems, as the problem with chaotic systems is that even their perfect solutions vary wildly with the starting conditions.
Fantastic . !
I love chaos theory. Its kinda my obsession. Thank you for a great video.
Finally somebody who said it. The video is on point and cannot be stressed enough. This type of mathematical philosophical thinking should be ingrained in our teaching
Great video
Thanks for the video, and thanks again for the well thought comments.
Not only is the decimal point dependent on the precision of the method and equipment used to measure it. The amount of times you measure it affects the 'confidence' of the accuracy of the measurement.
This is the entire idea of standard deviations and such. All measurements have to come with the 'range' of measurements that it is likely to be within. To be really 'scientific'.
I love the idea of chaos, degrees of freedom, and super finite initial conditions. Such a great way to think about deterministic but chaotic systems.
Excellent video.
Sentence structure tough for yee
Great info but even greater T-shirt!❤
These sort of problems where output variables are then put back in as inputs to get a new output, and so on are hypersensitive to their initial condition. And by definition they are hypersensitive to the computational precision used to calculate the outputs. So when examining an equation or algorithm assess whether perturbation errors are compounded (multiplication, power) or diminished (division, roots) and it will indicate what kind of function you may have.
There cannot be an end to science. The more you learn, the more questions you have and the more we find out what is undiscovered!
5:30 - Did you just try and sneak in fractals in a math problem¿? No, OK this is where I draw the line...... 😩
😂
To think that science is done and we could ever know everything there is to know is wild and highly improbable. Truly discovering the quantum and subatomic worlds alone should keep us busy for centuries.
That's right! The key takeaway here is, even if something's not random, it might still be sensitive to initial conditions!
What a roundabout way to say small changes in direction have a big effect as distance traveled increases.
Can we just take a moment to appreciate how hard The Action Lab is working for us!! Definitely awesome. ❤❤
Agree 🗿🍷
How hard they are doing for us? Can we all take a moment and check our grammar?
@@billyphilly22 Check OUR grammar? Can we all take a moment and check how many individuals are here?
An n-body gravity simulation was one of the first pieces of software I wrote. Turns out there is a super efficient and accurate algorithm for this: the Velocity Verlet integration. Solutions are still chaotic though!
It was software that (arguably) first exposed chaotic mathematics. Ref 'Edward Lorenz'.
Often it's not as problematic, as it seems in theory, as there is often a range, that is considered to be effectively the same output. Often the relevant period of time is very short (Will it rain this afternoon, while we are out on our bikes?). When the first branches occure, that means a certain number of outcomes is possible, not everything in between. Like a tree you climb, the fine twigs are usually of no importance depending on your scale/size and you can focus on the few strong ones to choose the best...
Every end is a new beginning.
It's not just the inputs. The idea we can calculate equations with infinite precision is also bogus, meaning you can also get different results if you simply vary the precision of the calculation.
Yes, but the important point about chaotic system goes *much* deeper than that.
Anyone getting hung on the mechanics of the calculations may (?) have missed the deeper point.
@@JxH actually, it was a hypothesis: I'm fairly sure that doing nothing more than changing the precision of the calculation will induce the same chaos do which you are referring. Far from dismissing the chaos aspect, I was trying to add to it.
Of course. Just like one small decision can and often does change the course of the rest of your life. And that change in your trajectory will then affect the lives of countless others, and on and on...
Celestial butterfly effect 😊
0:39 - one of those columns (to the right) is the matrix falling text screensaver?
Awesome.
Can you explain the link between this and the Mandelbrot fractal bifurcations that show actually chaos create order and can be predicted but like you said if you get the right information and initial starting conditions but all so not all ways some enter these all most sin wave oscillation but not quite true sin eventually throwing it out and the more as u put it balls into the system they start reaching these harmonic waves that seem to fall in. And out of constructive destructive resonance with each other adding to an over all tone.
This is similar to the n-children problem.
If you have 1 or 2 children things are easy to predict and control but as soon as you have 3 kids in your home there is utter chaos.
Hi. I have a question. Five years ago you published a video showing the effect of a magnet on a mouse.Sry It's been a long time.
So Each cell is subject to a repulsive force.
c
Could we accelerate a living organism in this way without subjecting it to the unpleasant effects of increasing G?
the same effect can be achieved without varying the initial conditions. Simply vary the temporal resolution of the simulation (time interval between each calculation step) and watch it result in a different outcome very quickly.
5:00 I've graphed this before 😁
We computer science students call that: garbage in, garbage out. However great you algorithm is coded, if you put garbage data in you'll get garbage data out xD
Chaos isn't random, chaos is a ladder.
HAPPY NEW YEARS BOYS
Excellent😊😊😊😊😊😊
I'm glad you said "notion of time," because, you know, time doesn't exist. It's a human invention because of the cycles we experience.
Do the Mandelbrot set! This one was a good segway in to it!
Just to be annoying, Segway is a trademarked name for an electric transportation device. The word you're thinking of is segue, though they are pronounced the same way.
Thanks!
Than you!
There is also that computers can not do exact calculations (as we assume from above video), as they always calculate in discrete values that can not represent infinite fractions. In chaotic systems and iterative recalculations these "difference" will cumulate, "multiplate" or "exponate" their effects.
Example: In my youth I coded a simple planetary model (one star + one planet) just on newtonian motion. As the precision of the underlying numbers (number-types in the software) chosen was to coarse, the planet started to "gain energy" and spiraled out of the star-system within few years event though the input values for simulation were perfect for circular motion. This effect "multipated" also faster when the time-steps for the calculation was reduced.
Small changes in the input do not produce small changes in the output, which is the definition of non linearity.
Note as well that it’s not simply a matter of correctly defining the initial conditions in order to predict long-term behavior, because our models are never perfect either - eg in the planetary model, nothing but point masses in the void are considered.
chaos theory is mesmerizing
There are a few more reasons why it's tricky to predict where an object is going to be in the future, especially when it comes to smaller objects such as comets and asteroids. Like the assumption that the objects mass and the center of mass being constant. After a few passes by the sun an asteroid can lose a significant amount of mass.
The "end of science" is a receding goal. Every time we begin to understand a layer of reality, we discover a deeper layer. I think that's built into the nature of the universe.
A small note here from someone who was doing a work on that topic in University long ago. The chaotic behaviour occurs only in specific ranges of starting conditions (quite wide range). If you would check out the Poincaré section of the Hénon-Heiles system, you would notice that at certain value of energy the structure is not the same everywhere, and have its areas of stable motion, which are called resonances. So even if system may seem chaotic with most starting conditions, there still could be "islands" of stability there. It is called sometimes the "determined chaos" (if you run the simulation with exactly the same starting conditions, you will still get the same result, but in nature it is impossible to do anything absolutely precisely).
A noble quote I found in perhaps Discover or other monthly mag back in, like, 1983 attributed to "A professor of astronomy at a university in Florida" (risky to not be anon? Hmm.): "After devoting my life to studying the mysteries of the universe, I'd like to think Heaven will be a place where I can say, 'OK, I give up. What *was* the answer?' " A humble heart with a scientific mind.