Phase separation in a granular mixture starting from a chessboard configuration
Channel: Nils Berglund
Date: 10/10/2023

This simulation investigates a question asked in comments to the video https://youtu.be/DLs_xbOx81s , namely whether a granular mixture of two types of particles would show phase separation, even when starting from a chessboard configuration, in which particles of the two types alternate. The answer is positive. 
The simulation features particles of two different sizes and masses. The larger particles are 1.4 times as large, and 5 times as heavy as the smaller ones. The particles interact via a Lennard-Jones potential, and are subject to a viscous damping term, gravity directed to the right and slightly downwards, at a slope 1/10, and to a one-sided harmonic repulsion from the obstacle. There are periodic boundary conditions.  
This simulation has two parts, showing the evolution with two different color schemes:
Type of particle: 0:00
Initial x coordinate: 1:56
In the first part, the color of the particles depends on their type. In the second part, the particles' color depends on their x coordinate at the beginning of the simulation. 
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: 21 minutes 55 seconds
This is much faster than the previous similar simulation, as I discovered an improvement in the way particles are drawn - in fact, many previous sims of LJ particles drew them as circles made of an unnecessarily large number of segments.

Color scheme: Part 1 - Turbo, by Anton Mikhailov
https://gist.github.com/mikhailov-work/6a308c20e494d9e0ccc29036b28faa7a
Part 2 - HSL/Jet

Music: "Red Eye" by E's Jammy Jams@ethaneubanks749

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)

#granular_material #lennardjones #LennardJones #molecular_dynamics #phonon