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October 18

Particle Collisions That Never Leave A Bounded Area

Yes. Two. See Orbit for more bounded solutions that don't collide. DroneB (talk) 13:56, 18 October 2018 (UTC)[reply]

This question came to me while trying to fall asleep: is there a set of identical balls of given radius and mass that obey classical physics, so that, for some initial position and velocity, will never leave a bounded area due to collisions within the set? If yes, what is the smallest number of balls needed - and would all larger sets of balls admit some such initial condition (is there a sequence of sets so that each one is a subset of every larger and so each stay a in a bounded area)? If yes, how does the diameter and shape of the bounds evolve with numbers of balls? Is there any scaling relation between radius from center of mass and velocity that will preserve boundedness? If not, how would you prove this?Phoenixia1177 (talk) 04:57, 18 October 2018 (UTC)[reply]

I think it is not possible (assuming classical physics means elastic collision), but one edge case escapes me, and I probably used too much math behind the scenes.

To simplify, assume the bounded area is the interior of a sphere (take any sphere that includes the area you want to use). Assume there is one such set of trajectories. Consider the center of mass of the set of the balls. It is moving at constant speed, so it will eventually leave the bounded area if that speed is not zero, and then at least one of the balls is outside the bounded area (our sphere is convex). Let's say this center of mass is located at the origin.

For any ball, call "outgoing speed" the scalar product of its speed with its normalized position vector (i.e.

{\overrightarrow {\partial _{t}r}}\cdot {\overrightarrow {r}}/r

with r the radius in spherical coordinates). Considering the outmost ball A (i.e. the furthest away from the origin) at some point. Without collisions, its outgoing speed does not decrease (that sounds obvious but really isn't, because of spherical coordinates tricks; for instance, it can increase e.g. imagine the ball passing near the origin, its outgoing distance goes from negative to positive). The only possible way its outgoing speed eventually decreases is by collision from another ball B that is further away from the origin. In order for that to happen, we need B to have a higher outgoing speed that A just before the collision (else it doesn't catch up) and obviously B is outmost-er than A before collision. The collision can only increase B's outgoing speed, thus at that point B is both further away and at a higher outgoing speed than A was. In summary, the outgoing speed of the outmost ball never decreases during time.

If that speed is nonnegative at any point, it's over: wait (diameter of bounding area)/(outgoing speed) time and the outmost ball is outside the bounding area. But "always increases" does not mean it eventually becomes nonnegative. It would be conceivably possible that the outmost ball always comes toward the origin, at lower and lower speeds. Intuitively it is kind of shocking (this requires the inside balls to never escape to another axis, yet give the bounding ball "slowing down" pushes that get more and more gentle as time passes) but I do not see a proof that it is impossible. Tigraan^{Click here to contact me} 08:21, 18 October 2018 (UTC)[reply]

You probably want to specify that at least one of the initial velocities is non-zero, otherwise it's trivially true. CodeTalker (talk) 13:47, 18 October 2018 (UTC)[reply]

If classical physics requires the balls to have mass, finite velocities and gravity, there is always a finite limit to how far they can separate. DroneB (talk) 14:20, 18 October 2018 (UTC)[reply]

@DroneB: You correctly point out that "classical physics" may imply long-distance forces between the balls (in which case my argument above collapses). However, it is not true that a set of gravity-attracted masses will always have a bounded maximum diameter. For instance, in the one-body problem, there are unbound orbits (see "orbit families" in [1], anything with an initial speed above the escape velocity is unbound). Tigraan^{Click here to contact me} 16:40, 18 October 2018 (UTC)[reply]

@Tigraan: Thank you for your correction. DroneB (talk) 17:00, 18 October 2018 (UTC)[reply]

I think that the answer is no as this question is equivalent to asking if one can create a spatial finite configuration of gas molecules that will never expand beyond some pre-defined limits. From what is known about kinetics of gasses it is not possible. Ruslik_Zero 20:27, 18 October 2018 (UTC)[reply]

When was the last time Crimea was an island?

Sagittarian Milky Way (talk) 13:55, 18 October 2018 (UTC)[reply]

It may have been an island during the Eemian interglacial, but that's contested [2]. Mikenorton (talk) 14:28, 18 October 2018 (UTC)[reply]

Although this study seems pretty sure that it was. Mikenorton (talk) 15:03, 18 October 2018 (UTC)[reply]