One thousand balls dropped on a double well curve (~ x^4

Re:One thousand balls dropped on a double well curve (~ x^4 ... by tnecniv on Tuesday June 20, 2023 @03:01PM (#36408873)

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.

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Re:One thousand balls dropped on a double well curve (~ x^4 ... by superposeur on Tuesday June 27, 2023 @04:19PM (#36498111)

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 :) )

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