How irreversible is the effect of a blender in a granular medium?
Channel: Nils Berglund
Date: 01/10/2023

In comments to the simulation https://youtu.be/cUUKPMSZXDY of a rotating triangle in a granular medium, the question arose to what extent this rotation is reversible. If the triangle were to rotate back to its initial position, how different would the particles' final position be different from their initial position?
This simulation investigates the question in the case of a "blender", composed of three blades angled at 120 degrees. The blender starts at rest, to give the medium a chance to reach an equilibrium configuration. Then the blender rotates 360° in one direction, before rotating back to its initial position. The angle is given in degrees, and the angular speed in radians per simulation second. It turns out that while particles near the circle described by the tips of the blender don't generally move back to their initial position, those closer to the center pretty much do. 
The particles in this simulation interact via a Lennard-Jones potential, and are subject to a viscous damping term and to a one-sided harmonic repulsion from the blender. There are periodic boundary conditions. 
This simulation has two parts, showing the evolution with two different color schemes:
Initial y coordinate: 0:00
Direction of velocity: 2:12
 In the first part, each particle is given a color at the beginning of the simulation, which is a function of its y-coordinate. The particle then keeps the same color throughout the simulation.In the second part, the hue of the particles depends on the direction of their velocity, and the luminosity depends on their speed. 
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: 24 minutes
Color scheme: Part 1 - HSV/Jet
Part 2 - Twilight by Bastian Bechtold
https://github.com/bastibe/twilight

Music: Enchanted Journey by Kevin MacLeod is licensed under a Creative Commons Attribution 4.0 licence. https://creativecommons.org/licenses/by/4.0/
Source: http://incompetech.com/music/royalty-free/index.html?isrc=USUAN1100799
Artist: http://incompetech.com/

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