This is the second installment of a new spin-off series, based on some "Mangroves vs tsunamis" simulations. It shows 1542 particles in a rectangular enclosure, interacting with a Lennard-Jones potential and subject to viscous damping. The color of the particles represents their kinetic energy, which is proportional to the square of their speed. Red particles move faster than blue ones. As time goes on, the overall energy decreases due to the damping, and the particles start forming a triangular lattice. Due to the rectangular boundary, however, a perfect lattice cannot form, and lines of crystallographic defects (grain boundaries) between regular regions remain, which tend to reorganize themselves in a series of "avalanches".
The Lennard-Jones potential is strongly repulsive at short distance, and mildly attracting at long distance. It is widely used as a simple yet realistic model for the motion of electrically neutral molecules. The force results from the repulsion between electrons due to Pauli's exclusion principle, while the attractive part is a more subtle effect appearing in a multipole expansion. For more details, see https://en.wikipedia.org/wiki/Lennard-Jones_potential
Statistical physics provides us of a rather good understanding of the dynamics of gases (ideal and non-ideal ones such as the Van der Waals gas), and of crystal lattices. Some phase transitions are also rather well understood, for instance why magnets lose their magnetization when heated. It is however much more difficult to model transitions between gaseous and solid phases, when freely moving particles tend to form regular arrangements when cooled down.
This simulation is inspired by discussions with Florian Theil, who proved a lot of difficult mathematical theorems on such systems: https://arxiv.org/search/?searchtype=author&query=Theil%2C+F
The motion of the particles is simulated using a Verlet algorithm. 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. Rather than using viscous damping, it would be more realistic to use a so-called thermostat to tune the temperature of the system. These will be used in some forthcoming simulations.
Render time: 4 minutes 20 seconds
Color scheme: Turbo, by Anton Mikhailov
https://gist.github.com/mikhailov-work/6a308c20e494d9e0ccc29036b28faa7a
Music: Gnarled Situation by Kevin MacLeod is licensed under a Creative Commons Attribution 4.0 licence. https://creativecommons.org/licenses/by/4.0/
Source: http://incompetech.com/music/royalty-free/index.html?isrc=USUAN1100405
Artist: http://incompetech.com/
Current version of the C code used to make these animations: https://github.com/nilsberglund-orleans/YouTube-simulations
https://www.idpoisson.fr/berglund/software.html
Some outreach articles on mathematics:
https://images.math.cnrs.fr/_Berglund-Nils-1343_.html
(in French, some with a Spanish translation)
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