Let it snow: Interface growth with molecular dynamics
Channel: Nils Berglund
Date: 2021-12-28

In this molecular dynamics simulation, the particles at the bottom of the display act as "seed" for a growing crystal. In order to keep them from moving much, they have been given a mass that is 10 000 times as large as the mass of the other particles . In addition, a little gravity has been added to make particles drift downward, new particles are added at random places and regular times at the top of the display, and the temperature is slowly decreased. As particles tend to pile up at the bottom, the pressure on the lower layers of particles slowly increases, thereby causing the arrangement to become more ordered.  
The resulting behavior can be viewed as a simple model for epitaxial growh of a crystal by deposition of atoms, see https://en.wikipedia.org/wiki/Epitaxy Some of these growth models can be described by the so-called KPZ equation, introduced by Mehran Kardar, Giorgio Parisi, and Yi-Cheng Zhang in 1986: https://en.wikipedia.org/wiki/Kardar%E2%80%93Parisi%E2%80%93Zhang_equation - see https://youtu.be/sQQyvIdW2sc for another type of interface growth simulation on this channel. 
The color of the particles represents the number of neighbors within a fixed distance. Blue particles have more neighbors than cyan, yellow, orange and red ones. There are two parts in this simulation. In the first one, only the particles are displayed, while in the second part, neighboring particles are connected by line segments that can be viewed as hydrogen bonds.
Without bonds: 0:00
With bonds: 2:25
The temperature is controlled by a thermostat, implemented here with 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

Render time: 26 minutes
Color scheme: Turbo, by Anton Mikhailov
https://gist.github.com/mikhailov-work/6a308c20e494d9e0ccc29036b28faa7a

Music: "Hurts So Good Blues" by the Unicorn Heads@UnicornHeads

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)