Crystal formation in a Lennard-Jones interacting particle system with a  thermostat
Channel: Nils Berglund
Date: 2021-12-09

This new simulation of particles interacting with a Lennard-Jones potential uses a thermostat instead of viscous friction, which allows to control the temperature of the system. 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 the temperature decreases, the average kinetic energy decreases as well, and particles tend to arrange into a regular crystal, though the rectangular boundary does not allow them to form a perfect triangular lattice.
The thermostat used here is 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 paticles 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, which represents the average kinetic energy of the particles. 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
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. 

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

Render time: 4 minutes 10 seconds
Color scheme: Turbo, by Anton Mikhailov
https://gist.github.com/mikhailov-work/6a308c20e494d9e0ccc29036b28faa7a

Music: "Rewired" by The Whole Other@thewholeother

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)