Is the billiard in a stadium reversible?
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- čas přidán 25. 04. 2021
- 200 non-interacting particles in a stadium-shaped domain. Their color changes at each reflection on the boundary. Leonid Bunimovich has shown that their dynamics is chaotic (ergodic and mixing). But is it really reversible, as it should be, according to the laws of classical dynamics? Let's find out...
Reversible means here that reversing the velocities of all particles (rotating their direction by 180° but keeping their speed) should be equivalent to changing the direction of time.
For context, see Richard Feynman's lectures on entropy: • Richard Feynman's Lect...
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This is a higher-quality remake of the video • Billiard in a stadium that has become a casualty of the CZcams compression algorithm.
Current version of the C code used to make these animations: github.com/nilsberglund-orlea...
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There just had to be that one little bastard at the end who couldn’t stay in the pack.
That's right, probably a round-off error. At least it shows I didn't cheat doing the simulation!
@@NilsBerglund Yeah I guessed there would be some rounding issues. This is a really good demonstration still though.
Typical, he had just one job.
@@NilsBerglund yes it's nice to see you reversing it at the end so they all go back to where they went, could you do it more often?
@@NilsBerglund Absolutely rounding errors. Double precision calculations are just not accurate enough. Probably need at least quadruple precision.
that one little sneaky particle to the right at the end...
I call it Brian, because of the Pythons...
And the moral of the story is, if you're moving in any direction and instantaneously turn 180° whilst maintaining speed you will travel back in time
No but you'lll fall over and bang your head so hard that you will think you are travelling back in time.
Assuming your entire body doesn't turn into red confetti from the sheer centrifugal forces it'd experience from instantaneously turning 180° on the spot... then again, I'm probably just overthinking the situation.
This is why I enter CZcams
I think it's interesting how quickly it's apparent that the mixing is reversed, our minds must be attuned to subtle trends in the behaviors of systems such as this.
Yeah I noticed the same thing, right away, like, something is off....
Our brains are hard wired to find patterns, and are extremely good at it.
I love that you are not only able to make these really cool simulations, but you're also able to explain, to a degree, how and why the simulations work. I'm normally not interested in things like this, but seeing you respond to common questions in the comments is actually quite informative.
Thanks! I believe that quite a few important concepts of physics and mathematics can be made easier to grasp by appropriate graphical illustrations - at least to people with an eye for such things. It's cool that it seems to work.
Reminds me of reversible cellular automata like critters. Good content.
Thanks!
What cellular automata
@@zeljkarozman3084 critters
@@zeljkarozman3084 nanobots
Me:*sees the title* what
Also me:haha funny points go brr
Nils has heard our prayers. Praise Nils.
I gave you your 69 likes
I appreciate this being done correctly, rather than the second half just being the first half played backwards.
oddly fascinating despite any other outcome not making sense. any reason you can think of as to why most but not all particles ended up red at the end?
That's probably because they have all made about the same number of collisions with the boundary. That would change if I let the simulation run for a longer time, though at some point round-off errors destroy reversibility.
Nils Berglund I find that pretty interesting
Would something interresting happen if you rotate them by 270°? Maybe a time of i? i as sqrt(-1)
You mean changing the reflection rule? I'm not sure right now which one would give interesting dynamics, but I can look into it.
@@NilsBerglund I think they meant, rotating 270° instead of 180 like you did. Idk what it’s supposed to do either.
I see. Most probably, it would keep looking very random. The point is, the configuration of the particles when I briefly stop the animation in the middle of the video is quite disordered (it has a large entropy), and only by reversing velocities in this very particular way will you be able to return to the ordered state.
I did mean whar EnTi Zed said. What I meant is that if rotating it 180° means multiplying the flow of time by -1, rotating it by 270° could mean multiplying it by i. I didn't think it would do anything interesting or that even the idea was correct, but I found the idea of a time on the complex plane, almost as if it were two-dimensional, interesting
This is the best illustration of the second law I've seen.
Thanks for sharing!
Thanks for watching!
I was honestly not expecting that. My intuition has been wrong before but never in such a satisfactory manner.
That makes me curious, is there a non trivial position (say, all rays aim at the perpendicular of the shape) that can return to its original position? Something that would become chaotic but return to order?
Edit: I saw the response for another comment saying that if you wait long enough you can get as close as you want to your original position.
So it begs two questions: can we approximate how long? And, is there a guarantee to reach the same position, or just a limit of it?
I dont't know if you can estimate the recurrence time very precisely. But there are simplified model, such as the Ehrenfest urn model (look for it on Wikipedia, or here for an impression: czcams.com/video/U_XOy9GtfGw/video.html ) allows to give an estimate - the time is exponentially long in the number of distinguishable states. And regarding your second question, it will only be an approximation. The closer you want to get, the longer you have to wait.
I'm so happy for them!!
I like your simulations
Thanks, I'm glad you do!
This is awesome! What would make it nonreversible would be to add coupling inter-billiard force terms. That would be a cool demonstration of entropy!
Coupling the particles by reversible forces would leave the system reversible. This is very nicely explained by Richard Feynman in the talk linked to in the video description.
@@NilsBerglund Right, I guess what I meant was that it would seem ergodic in the time scale of the video.
Is there any initial state that, if run long enough will converge again?
Not including straight up or directly sideways
Maybe sideways would be cool
Poincaré's recurrence theorem says that if you wait long enough, you will come as close as you like to the initial state. However, long enough may mean very, very long...
The focis
@@NilsBerglund Is the simulation discrete?
Congratulation! You made the billiard science into the bullet hell room
The out of order colors at the end confuses me!
Did they mostly return yet encoded with information about the shape they were in?
Or is the miss match the difference between the central release point and the larger return point (and thus if the calculations more accurate, the colors would line up better too)?
The color of each particle changes with each collision with the boundary. The colors at the end are different because the particles have not all made the same number of collisions.
If you would not stop after reaching the origin again but let it run into "negative time", do you think there would be some kind of symmetry emerging? I cannot think of an obvious relationship, but it would be interesting to see.
In general, I don't see an obvious symmetry either. Unless the initial condition is itself symmetric, for instance with respect to one of the symmetry axes of the stadium.
Impossibruuuu!! What precision do you work with in your calculations? Shouldn't the results go into a chaos at reverse?
I use double-precision floating point. You're right that if I were to run it much longer, the round-off errors would prevent it from returning to the initial state.
Reversible ? Yes. The disparity at the end is only caused by computational errors from the limited granularity of the FPU. In this case, 64 bits of precision is just not enough. Need about double. We desperately need 128 bits, quad precision, FPUs, or even 256 bits ones, octuple precision.
I imagine if there was friction involved with the walls, the system would be non reversible?
That's right, because energy would be lost to the walls. The same would hold for viscous drag inside the billiard domain, if this were filled with some kind of fluid.
I dunno about you but I just put Touhou OST over this and it was actually fitting for how chaotic this becomes.
Just like that, I imagine that the cycles of nature and existence work.
Now what if they interacted (bounced off each other like they were walls)?
It would not change the dynamics, because all particles have the same mass and velocity, so bouncing off each other is the same as just crossing each other. For more complicated interactions between the particles, the simulation would probably become much more sensitive to round-off errors.
any bullet hell game be like:
enthropy always goes up
this is both intuitive and very confusing, because this is one of the tricks your brain uses to know if the video is in reverse or not. When doesn't follow this rule, something feels off, yet nothing seems impossible when observed carefully.
enthropy is just the numbers of posible microstates that a macrostate has, and, because of statistics, there are more ways to put all those lines at random than to have them all in the same place, so is more likely to have them at random than to have them synchronized, not because it's impossible, just by mere chance, because it's a more likely scenario (that's why this is not actually a law of enthropy, but our brain is made to think that is a law, and it always works)
Ludwig Boltzmann couldn't have said it better.
Under ideal theoretical conditions, entropy is reversible. Damn.
this is interesting
I would say that so long as only geometric interaction occurs, it will always be reversable.
could you show how billiard in doughnut look like
You could try how reflections would work inside of company logos?
Anything but company logos.
@@OblivionFalls but y?
shouldn't any system like this that progresses with time always be reversible, even if it behaves chaotically? (ignoring very fine rounding errors)
Yes, that's correct.
There's entropy that prevents that
it gets increasingly hard to follow just one
Woah!!!
Is there a situation where it would not be reversible?
If you add a magnetic field, it would only be reversible if you change the direction of the field when reversing velocities. Another way of making the system non-reversible would be to add randomness.
Holy crap, man. The fact that the entire thing returns to the intial state by simply inverting the velocities at a random point in the simulation baffles my monkey brain.
Well, it is pretty intuitive if you think about it - inverting the velocities was simply making them trace the very same path they went beforehand, nothing more. Infact, it was almost the same as playing the video in reverse!
@@yoavmal I added a link to a lecture by Richard Feynman in the description, for more context.
@@yoavmal I kinda realize that now. I'm just rather slow.
Why would any Billard simulation not be reversable?
You're right, of course, that it should be. Except for round-off errors.
Why is it reversible?
Because Newton's laws of dynamics are. You can also see it by considering each individual particle, moving at constant speed on straight lines with occasional changes of direction.
@@NilsBerglund On the start of this video, you'll see that the billiard lines are perfectly placed with carefulness. And, in the end of this video, what will happen if the billiard lines are imperfectly placed with awareness?
@@wagnerramosmidichannelabso514 Not sure what you mean there.
Daamn
Mmm sprinkles
Ants!
Goose
Why is the stadium so chaotic though?
It has to do with a "focussing-defocussing" mechanism. Particles reflected on one of the circular arcs tend to be focussed towards the arc's center, but when traveling to the other arc, they tend to spread out even more.
@@NilsBerglund ah
looks like a buncha bugs walkin around (:
I here the music, but I do not see jobbythehong transforming soemthing 0/10
I hope to all that is holy someone gets that's a joke
Swivel here
Why are these on my recommended now
Jobby fans where y'all at?
а... эммм... чё? wtf is this?
gauss
This is what happened when the Doctor reversed the polarity of the neutron flow
Yeah, TARDIS probably has to mess with entropy somehow...
You know, people think I'm smart enough to understand shit like this, but ever since I came to realization that I'm a complete dumbass I don't so, what's the point of this experiment? Why wouldn't it go back to it's original state if it was essentially reversed?
In a way, it's not surprising at all, you're right. It is just that to us, the state in the middle of the animation looks very random (it has a high entropy), so if we were to start from there, it would look a bit like a heap of trash spontaneously assembling to become a Boeing 747, or something.