If 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?
Quoting 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.”
Right. 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.
They 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.
Why 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.
The 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.
This makes sense, but then again my question is motivated by the astronomical example, which still seems mysterious.
Naively the various arguments and theorems about thermal equilibrium (at least macroscopic equilibrium as you point out) would seem to apply to the situation of 10^24 bodies bouncing around and interacting gravitationally (with some feedback mechanisms into other modes)… basically like a box of billiard balls interacting mostly electromagnetically.
Yet there is no “gravitational temperature” that allows one to predict the expected speed of a rock of a certain mass floating in the solar system. (Whereas knowing the temperature of a gas in a box gives good statistical information about the speed of a molecule of a certain mass in the box.)
Incidentally I’ve put some version of this question to four different physicists and gotten four different replies! (“Chaos” being one of these :) )
I think a lot of the answer is 'chaos'. Most nontrivial systems are chaotic (generically, iirc more than 3 unconstrained d.o.f. + non-linearity) and most chaotic systems quickly erase knowledge of the initial conditions (https://en.wikipedia.org/wiki/Lyapunov_exponent). That being said, within chaos, there are fractal regions that survive pertubations (https://en.wikipedia.org/wiki/Kolmogorov%E2%80%93Arnold%E2%8...) and this is indeed related to why the solar system is more stable and predictable (can read about KAM theorem and Jupiter).
In the bouncing ball example, there are periodic orbits that are created for some balls, in which case you can perfectly predict their past and future. But a lot of measurements we do make about the world tend to be statistical measurements that don't depend on measuring the precise trajectory of individual particles over long periods of time. So these issues tend to not be important unless precise details are needed.
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[ 3.3 ms ] story [ 36.4 ms ] threadHere's the link to the right time ... the 130 second mark:
https://www.youtube.com/watch?v=LQmyR-sC6GA#t=130s
> In the 1000-ball part balls are initially mutually separated by less than a millionth of the plot width.
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.
There's also similar work framed in terms of billiards:
https://en.m.wikipedia.org/wiki/Dynamical_billiards
[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.
Naively the various arguments and theorems about thermal equilibrium (at least macroscopic equilibrium as you point out) would seem to apply to the situation of 10^24 bodies bouncing around and interacting gravitationally (with some feedback mechanisms into other modes)… basically like a box of billiard balls interacting mostly electromagnetically.
Yet there is no “gravitational temperature” that allows one to predict the expected speed of a rock of a certain mass floating in the solar system. (Whereas knowing the temperature of a gas in a box gives good statistical information about the speed of a molecule of a certain mass in the box.)
Incidentally I’ve put some version of this question to four different physicists and gotten four different replies! (“Chaos” being one of these :) )
In the bouncing ball example, there are periodic orbits that are created for some balls, in which case you can perfectly predict their past and future. But a lot of measurements we do make about the world tend to be statistical measurements that don't depend on measuring the precise trajectory of individual particles over long periods of time. So these issues tend to not be important unless precise details are needed.