3 ms·
One thousand balls dropped on a double well curve (~ x^4 – x^2) [video]
- ColinWright 3y agoThe original URL provided linked to the time when the 1000 balls were dropped, but the submission process lost the time-stamp. Here's the link to the right time ... the 130 second mark: https://www.youtube.com/watch?v=LQmyR-sC6GA#t=130s https://www.youtube.com/watch?v=LQmyR-sC6GA#t=130s
- Xophmeister 3y agoIf the balls don’t interact with each other and they all have the same initial condition, why would their paths diverge? Is it just down to lack of numerical precision, that compounds on each bounce?
- whenlambo 3y agoQuoting the author: “Whenever I animate more than one ball in these animations, the starting positions are slightly different. For chaotic systems, like this, the distances between the balls increases exponentially.”
- rubatuga 3y agoVideo description: "In the 1000-ball part balls are initially mutually separated by less than a millionth of the plot width. "
- pavel_lishin 3y agoI was wondering this too! The video description states: > In the 1000-ball part balls are initially mutually separated by less than a millionth of the plot width.
- vikingerik 3y agoRight. Also I think the initial drop position is deliberately selected to get the balls separating sooner. Once they get over the hump to the right side, they make several successive bounces at very shallow angles to the surface, which will magnify a small difference in ball position into a greater difference in bounce position and timing.
- Macha 3y agoThey don't have the same initial condition. It appears that in the rendering because it has a much lower resolution than the simulation it is rendering.
- superposeur 3y agoWhy is it that some systems come to equilibrium quickly, effectively erasing the imprint of initial conditions, while others do not? For instance, if we look at the distribution of astronomical bodies in the universe, almost nothing can be understood without knowing the initial conditions. Or, to take the other side of the coin: by studying the present day state of astronomical bodies, we can infer quite a bit about the initial conditions of the universe. Whereas, knowing a later state of the balls in the video tells us almost nothing about where the initial ball drop occurred. Perhaps the answer is “chaotic dynamics”. Yet gravitational interactions can certainly be chaotic. And some chaotic systems still don’t “come to equilibrium”. I suppose I’m asking how one might look at the dynamical laws at a glance and say one system will come to equilibrium and another will not.
- mrfox321 3y agoI'm not in the field, but some of your questions are research problems regarding boundary conditions of the laplacian operator. There's also similar work framed in terms of billiards: https://en.m.wikipedia.org/wiki/Dynamical_billiards https://en.m.wikipedia.org/wiki/Dynamical_billiards
- tnecniv 3y agoThe answer really is chaos. Chaos has different definitions used by different authors, but a characteristic they all capture is sensitivity to initial conditions. This requires inherently unstable dynamics (convergence to a fixed point or limit cycle corresponds to an insensitivity to initial conditions) It’s also important to distinguish between microscopic and macroscopic equilibrium. The system shown in this video seems, at least informally, chaotic at the microscopic level (the dynamics of each particle). However, if you consider the dynamics of the distribution (i.e., the ensemble of a very large, ideally infinite) number of balls, it is well known that this distribution converges to an equilibrium distribution known as the Gibbs measure [1]. [1] I’m on my phone and not thinking too hard, you may need to add some Brownian noise to the dynamics for this experiment, but the double-well is commonly used to demonstrate this behavior.
- superposeur 3y ago