Spherical pool with pockets in deeper depressions

Описание: This is a variant of the simulation    • Spherical pool with pockets inside depress...   , showing spherical pool in which the pockets are surrounded by "depressions". The depressions are modeled by rotation-symmetric potentials centered in the pockets, that exert a central force on the particles, as if the pockets were attracting. Compared to the previous video, the force constant is 10 times larger, so that the influence of the depressions on the trajectories of the balls is more visible. The pockets have also been made much smaller, to have a better chance to see balls orbiting a pocket instead of falling into it.
Note that this is different from the situation where the particles follow geodesics on a deformed sphere, which would require computing the deformed metric, but the result should not be very different. In fact, in the spirit of general relativity, there should exist a specific deformation compatible with the observed trajectories. Note, however, that the deformation would depend on the particles' momentum.
The incoming particle is shot at a set of immobile particles, in a similar configuration to what one would do for pool billiard, but on a sphere. There is no friction acting on the particles, and also no thermostat. The motion of the particles is governed solely by a Lennard-Jones interaction between them and the force deriving from the potential.
The video has two parts, showing the same simulation with two different representations:
3D view: 0:00
2D view: 1:28
In both parts, the color of the particles depends on their kinetic energy. The 2D part shows an equirectangular projection, meaning that the x- and y-coordinates are proportional to the longitude and latitude of the particles. Particles move in apparently curved lines due to the projection - you see similar paths for spacecraft and satellites orbiting the Earth. Particles should actually have elongated elliptical shapes when approaching the poles, but we chose not to do this here. This is also why atoms of the same molecule can appear to be far from each other near the poles. In the 3D parts, the observer moves around the sphere in a plane containing the center of the sphere. The number of particles that have fallen into pockets over time is shown at the top right.
To save on computation time, particles are placed into a "hash grid", each cell of which contains between 3 and 10 particles. Then only the influence of other particles in the same or neighboring cells is taken into account for each particle.

Render time: 3D part: 16 minutes 55 seconds
2D part: 3 minutes 3 seconds
Compression: crf 28
ffmpeg added noise option: -vf noise=alls=10:allf=t+u
Color scheme: Turbo, by Anton Mikhailov
https://gist.github.com/mikhailov-wor...

Music: "Nobody Calls it San Fran" by Coyote Hearing‪@felte‬

Current version of the C code used to make these animations:
https://github.com/nilsberglund-orlea...
https://www.idpoisson.fr/berglund/sof...
Some outreach articles on mathematics:
https://images.math.cnrs.fr/auteurs/n...
(in French, some with a Spanish translation)

#particles #sphere #LennardJones