Not quite a quasicrystal (yet): Particles interacting with a potential based on the golden ratio
Channel: Nils Berglund
Date: 2022-01-05

This new attempt at growing a quasicrystal uses a potential with rotation symmetry, which is similar to the Lennard-Jones potential, but has two stable equilibrium positions at distances whose ratio ls the golden mean Phi = 1.618.... This was suggested to me by Florian Theil. The result is not yet quite convincing, though one does observe a predominance of cyan particles, which have 5 neighbors, at a certain point in the simulation. In the end, however, the atoms seem to prefer a triangular lattice and the crystal tends to shrink. These simulations appear to be very sensitive to parameters in the potential and in the cooling/heating schedule, so it is quite possible better results can be obtained with other parameter values. In the meantime, I have also been experimenting with simulations featuring two different kind of particles, as suggested by several viewers. Some of them will be posted in a few days.   
The exact form of the potential in polar coordinates is 
V(r) = (1/12)*(req/r)^12 - (b/9)*(req/r)^9 + (c/6)*(req/r)^6 - (d/3)*(req/r)^3
where req is 5 times the radius of the particles, and the parameters are b = 1 + Phi + a, c = Phi + a + a*Phi, and d = a*Phi. The parameter a, which controls the location of the local maximum of the potential, as well as the depth of the two potential minima, has been set to 1.1.
The color of the particles represents the number of neighbors they are connected to by a link. Blue particles have more neighbors than cyan, yellow, orange and red ones. Temperature is controlled by a thermostat, given here by the "Nosé-Hoover-Langevin" algorithm introduced by Ben Leimkuhler,  Emad Noorizadeh and Florian Theil, see reference below. The idea of the algorithm is to couple the momenta of the system to a single random process, which fluctuates around a temperature-dependent mean value. Lower temperatures lead to lower mean values. The temperature shown on the top right is in fact the temperature multiplied by Boltzmann's constant. 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. 
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

Edit: After an email exchange with Florian Theil, I realized I should replace Phi by its the inverse of its third power Phi^3 in the above potential. If this gives better results, I will post that simulation in a few days.

Render time: 34 minutes
Color scheme: Turbo, by Anton Mikhailov
https://gist.github.com/mikhailov-work/6a308c20e494d9e0ccc29036b28faa7a

Music: "Airline" by Geographer@geographermusic

Reference: Leimkuhler, B., Noorizadeh, E. & Theil, F. A Gentle Stochastic Thermostat for Molecular Dynamics. J Stat Phys 135, 261–277 (2009). https://doi.org/10.1007/s10955-009-9734-0
http://www.maths.warwick.ac.uk/~theil/HL12-3-2009.pdf

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)