Two types of particles interacting with Lennard-Jones-type potentials depending on the golden ratio
Channel: Nils Berglund
Date: 21/01/2022

Like the simulation https://youtu.be/lnlZlq1owYQ this one shows the evolution of a mixture of two types of particles interacting with a radial potential. Here there are 1495 particles, 80% of which are of type I (indicated by small circles), and 20% of which are of type II (indicated by larger circles marked with an x). 
Particles of type I create a rotation symmetric force field with two different stable positions, at distance 5*r and 5*Phi*r, where r is the radius of the circles, and Phi = 1.618... is the golden mean. The potential this force derives from has been artificially constructed by patching together pieces of Lennard-Jones potentials, and of polynomials of degree 3.
Particles of type II interact with a standard Lennard-Jones potential, with equilibrium positions at distance 5*R, where R = Phi*r is the radius of the large circles. They also have a mass that is 10 times smaller than the mass of particles of type I.  
These choices of parameters were once again motivated by the hope to create something like a quasicrystal. The mass of the particles of type II has been chosen small because in the simulation https://youtu.be/lnlZlq1owYQ the particles with a larger equilibrium radius moved hardly at all, and indeed here they are more mobile. I don't think that the result can be called a quasicrystal, though I found the structure quite interesting. In particular, the quite stable hexagonal structures are somewhat reminiscent of zeolites. 
This video has two parts, showing the same evolution with two different representations:
Without atomic bonds: 0:00
With atomic bonds: 1:45
In both parts, the color of particles depends on the number of neighbors, defined as particles within a given distance. Blue particles have more neighbors than red ones. In the second part, particles considered as neighbors are connected by straight lines, that can be considered as hydrogen bonds. However, particles also interact when they are not connected by bonds. 
The temperature is controlled by a thermostat, which is here 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. 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

Render time: 19 minutes 20 seconds
Color scheme: Turbo, by Anton Mikhailov
https://gist.github.com/mikhailov-work/6a308c20e494d9e0ccc29036b28faa7a

Music:  Cool Ride by Audionautix is licensed under a Creative Commons Attribution 4.0 licence. https://creativecommons.org/licenses/by/4.0/
Artist: http://audionautix.com/

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)