Freezing and melting of particles with anisotropic Lennard-Jones interaction
Channel: Nils Berglund
Date: 2021-12-20

In this simulation, particles interact with an anisotropic version of the Lennard-Jones potential, which favors the horizontal, vertical, and diagonal directions. Any two particles, when far away from all other particles, would form a stable configuration when they are situated on vertices of a square lattice which are horizontal or vertical neighbors, or next-to-nearest neighbors along the diagonals. Unlike the usual Lennard-Jones potential, which is invariant under rotations, this potential favors the formation of a square lattice of particles as the temperature decreases. These configurations even remain rather stable when the temperature is increased again. Particles interacting with a non-isotropic potential do exist in nature, though it would probably be more realistic to introduce an angular degree of freedom for each particle, with respect to which the interaction is defined.  
The precise form of the potential in polar coordinates is 
V(r,phi) = (req/r)^12 - (9-7*cos(4*phi))*(req/r)^6, 
where req is equal to 5 times the radius of the particles. 
The color of the particles represents the number of neighbors, defined as particles within a fixed distance. 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
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. 

Render time: 17 minutes 30 seconds
Color scheme: Turbo, by Anton Mikhailov
https://gist.github.com/mikhailov-work/6a308c20e494d9e0ccc29036b28faa7a

Music: "Float" by Emily Sprague@emilysprague6983

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)