Phonons in a flowing granular material
Channel: Nils Berglund
Date: 27/07/2023

This is a variant of the video https://youtu.be/uLSCm33-lWY in which the density of particles has been decreased: it is approximately 83.6% of the density in the previous video. As a result, the particles are now able to flow in the direction of gravity, which is directed horizontally from left to right. The system thus appears to be below the jamming transition observed in granular materials. One can still see lattice defects propagate in the form of wave packets of phonons.  
This simulation has two parts, showing the evolution with two different color schemes:
Number of neighbors: 0:00
Averaged kinetic energy: 1:09
In part 1, the color of the particles depends on the number of neighbors within a fixed distance. Particles with fewer neighbors have a lighter hue. In part 2, the particle's color depends on their kinetic energy, but averaged over a time window, instead of the instantaneous kinetic energy shown in several previous videos. Red particles have a larger average speed than blue ones. This allows to visualize defects propagating through the lattice.
The particles are interacting via a Lennard-Jones potential, and with a one-sided harmonic force with the obstacles. They are subject to gravity, directed from left to right, and to a viscous friction. There are periodic boundary conditions.
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: 52 minutes 54 seconds
Color scheme: Part 1 - Inferno by Nathaniel J. Smith and Stefan van der Walt
https://github.com/BIDS/colormap
Part 2 - Turbo, by Anton Mikhailov
https://gist.github.com/mikhailov-work/6a308c20e494d9e0ccc29036b28faa7a

Music: "Positive Fuse" by French Fuse@frenchfuse

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